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AN ALGEBRAIC CHARACTERIZATION OF MINKOWSKI SPACE 



By 
RICHARD K. WHITE 



A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL 

OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT 

OF THE REQUIREMENTS FOR THE DEGREE OF 

DOCTOR OF PHILOSOPHY 



UNIVERSITY OF FLORIDA 



■V 



2001 



ACKNOWLEDGMENTS 



I thank my advisor Dr. Stephen J. Summers for his guidance in the 
preparation of this dissertation. I would also like to thank all of my committee 
members for their support and for serving on my committee. 



TABLE OF CONTENTS 

page 
ACKNOWLEDGMENTS ii 

ABSTRACT v 

CHAPTERS 

1 INTRODUCTION 1 

2 A CONSTRUCTION OF THREE-DIMENSIONAL MINKOWSKI SPACE .... 7 

2. 1 Preliminaries 9 

2.2 Construction of n 11 

2.3 Reflections About Exterior Points 18 

2.4 Embedding a Hyperbolic Projective-Metric Plane 22 

2.5 Exterior Point Reflections Generate Motions in an Affine Space 23 

2.6 Conclusion 24 

3 A CONSTRUCTION OF FOUR-DIMENSIONAL MINKOWSKI SPACE 26 

3.1 Preliminaries and General Theorems 26 

3.1.1 Properties oi M 30 

3.1.2 Properties of V 31 

3.1.3 Properties of X 32 

3.1.4 General Consequences of the Axioms 32 

3.1.5 Perpendicular Plane Theorems 33 

3.1.6 Parallel Planes 34 

3.1.7 Consequences of Axiom 11 and 3.1.6.18 37 

3.2 Lines and Planes 38 

3.2.1 General Theorems and Definitions 38 

3.2.2 Isotropic Lines 41 

3.3 A Reduction to Two Dimensions 44 

3.4 Consequences of Section 3.3 51 

3.5 Construction of the Field 52 

3.6 Dilations and the Construction of (3£, V,/C) 58 

3.7 Subspaces and Dimensions 66 

3.8 Orthogonality 69 

3.9 The Polarity 80 

3.10 Spacelike Planes and Their Reflections 86 



ni 



page 

4 AN EXAMPLE OF THE THREE-DIMENSIONAL MODEL 91 

5 CONCLUSION 96 

REFERENCES 103 

BIOGRAPHICAL SKETCH 106 



IV 



Abstract of Dissertation Presented to the Graduate School 

of the University of Florida in Partial Fulfillment of the 

Requirements for the Degree of Doctor of Philosophy 

AN ALGEBRAIC CHARACTERIZATION OF MINKOWSKI SPACE 

By 

Richard K. White 

May 2001 

Chairman: Dr. Stephen J. Summers 
Major Department: Mathematics 

We give an algebraic characterization of three-dimensional and four- 
dimensional Minkowski space. We construct both spaces from a set of involution 
elements and the group it generates. We then identify the elements of the original 
generating set with spacelike lines and their corresponding reflections in the three- 
dimensional case and with spacelike planes and their corresponding reflections in the 
four-dimensional case. Further, we explore the relationship between these 
characterizations and the condition of geometric modular action in algebraic 
quantum field theory. 



CHAPTER 1 
INTRODUCTION 



The research program which this dissertation is a part of started with a paper 
by Detlev Buchholz and Stephen J. Summers entitled "An Algebraic Character- 
ization of Vacuum States in Minkowski Space" [9|. In 1975, Bisognano and 
Wichmann [5] showed that for quantum theories satisfying the Wightman axioms 
the modular objects associated by Tomita-Takesaki theory to the vacuum state and 
local algebras generated by field operators with support in wedgelike spacetime 
regions in Minkowski space have geometrical meaning. Motivated by this work, 
Buchholz and Summers gave an algebraic characterization of vacuum states on nets 
of C*-algebras over Minkowski space and reconstructed the spacetime translations 
with the help of the modular structures associated with such states. Their result 
suggested that a "condition of geometric modular action" might hold in quantum 
field theories on a wider class of spacetime manifolds. 

To explain the abstract version of this condition, first some notation is 
introduced and the basic set-up is given. Let {Ai}iei be a collection of C*-algebras 
labeled by the elements of some index set / such that (/,<) is an ordered set and the 
property of isotony holds. That is, if /i,/2 e / such that /i < h, then A^ c Ai^. 
Let ^ be a C*-algebra containing {yl,}/^/. It is also required that the assignment 
/ 1-+ A, is an order-preserving bijection. 

In algebraic quantum field theory, the index set I is usually a collection of 
open subsets of an appropriate spacetime (M,g). In such a case, the algebra Ai is ^ 
interpreted as the C*-algebra generated by all the observables measured in a 



2 

spacetime region /. Hence, to different spacetime regions should correspond different 
algebras. 

Given a state co on A, let (7^(o,7t(o,Q) be the corresponding GNS 
representation and let TZj = 7tco(.A/)", / e I, be the von Neumann algebras generated 
by the 7T(o(^,), / e I. Assume that the map / >-^ Tlj is an order- preserving bijection 
and that the GNS vector Q is cyclic and separating for each algebra TZ,, i e /. From 
Tomita-Takesaki theory, we thus have a collection {J/},g/ of modular involutions 
and a collection {A/},g/ of modular operators directly derivable from the state and 
the algebras. Each J,- is an anti-linear involution on Jia such that JjTZiJi = TZi' and 
J,Q = Q. 

In addition, the set {J/}/6/ generates a group J" which becomes a topological 
group in the strong operator topology on B(Ti.io), the set of all bounded operators on 
Tiio. The modular operators {A,},g/ are positive (unbounded) invertible operators 
such that 

AfTZjAf = IZj, j ^ I, / e E, i = T-T and A''Q. = Q. 

In algebraic quantum field theory the state co models the preparations in the 
laboratory and the algebras Ai model the observables in the laboratory and are 
therefore, viewed as idealizations of operationally determined quantities. Since 
Tomita-Takesaki theory uniquely gives these modular objects corresponding to 
(TZj,Q.), it thus follows that these modular objects can be viewed as operationally 
determined. 

Motivated by the earlier work of Bisognano and Wichmann [5], Buchholz and 
Summers [9] proposed that physically interesting states could be selected by looking 
at those states which satisfied the condition of geometric modular action, CGMA. 
Given the structures indicated above, the pair ({^/},g/,co) satisfies the abstract 
version of the CGMA if {7^,■},■e/ is left invariant under the adjoint action of the 



3 

modular conjugations {J/l/g/; that is, if for every i,j in / there is a A: in / such that 

adJiiUj) = JiUjJi = Uk, where JiUjJi = {JjAJi : A e TZj}. 
Thus, for each / in /, there is an order-preserving bijection, automorphism, X/ on /, 
(/,<), such that J,TZjJi = Tlx,(j)- fory e /.The set {x,},-6/ is a set of involutions which 
generate a group T, which is a subgroup of the group of translations on /. Buchholz, 
Dreyer, Florig, and Summers [6] have shown that the groups Tarising in this 
manner satisfy certain structure properties, but for the purposes of this thesis, it is 
only emphasized that Tis generated by involutions and is hence, a Coxeter group. 

Thus there are two groups generated by involutions operating on two different 
levels. 

1. The group Tacting on the index set /. 

2. The group J7 acting on the set {7^/},e/. 

To elaborate further the relation between the groups Tand J^, consider the 
following. 

Proposition 1.1 [6] The surjective map £, : J^^T given by ^(J,, •■■J,-,„) = X/|---T/^, is 
a group homomorphism. Its kernel S lies in the center of J and the adjoint action of 
S leaves each 71 1 fixed. ■ 
Thus, J" is a central extension of the group Tby S. 

As an immediate consequence of this proposition, J" provides a projective 
representation of Twith coefficients in an abelian group Z in the center of J. Thus, 
the condition of geometric modular action induces a transformation group on the 
index set / and provides it with a projective representation. 

With this in mind, the following program was then posed. Given the 
operational data available from algebraic quantum field theory, can one determine 
the spacetime symmetries, the dimension of the spacetime, and the spacetime itself? 
That is to say, given a net of C*-algebras and a state co satisfying the CGMA, can 



4 

one determine the spacetime symmetries, the dimension of the spacetime, and even 
the spacetime itself? 

Part of this has been carried out by Buchholz, Dreyer, Florig, and Summers 
for Minkowski space and de Sitter space [6]. However, in order to do so, they had to 
presume the respective spacetime as a topological manifold. But would it not be 
possible to completely derive the spacetime from the operationally given data 
without any assumption about dimension or topology? 

As was pointed out by Dr. Summers, a possibility to do so was opened up in 
this program in the following manner. As already seen, the CGMA yields an 
involution generated group complete with a projective representation and there is in 
the literature a way of deriving spacetimes from such groups going under the name 
of absolute geometry. 

In general, absolute geometry refers to a geometry that includes both 
Euclidean and non-Euclidean geometry as special cases. Thus, one has a system of 
axioms not yet implying any decision about parallelism. In our case, the axioms are 
given in terms of a group of motions as an extension of Klein's Erlangen Program. A 
group of motions is defined as a set Q of involution elements closed under 
conjugation and the group it generates. In a group of motions the representations 
of geometric objects and relations depend only on the given multiplication for the 
group elements, without reference to any additional structure. The system of axioms 
is formulated in terms of the involutory generators alone, so that geometric concepts 
like point, line, and incidence no longer are primary but are derived. 

The necessary means for setting up this representation are provided by the 
totality of reflections in points, lines, and planes (a subset of the set of motions). 
Points, lines, and planes are in one-to-one correspondence with the reflections in 
them so that geometric relations among points, lines, and planes correspond to 
group-theoretic equations among the reflections. This enables one to be able to 



5 

formulate geometric theorems about elements of the group of motions and to be able 
to then prove these theorems by group-theoretic calculation. 

To summarize, we are to find conditions on an index set / and a corresponding 
net of C*-algebras {^/}/g/ as well as a state co satisfying the CGMA such that the 
elements of / can be naturally identified with open sets of Minkowski space and such 
that the group Tis implemented by the Poincare group on this Minkowski space. 
Out of the group Twe wish to construct Minkowski space such that T's natural 
action on Minkowski space is that of the Poincare group. 

This involves two steps. First we carry out the absolute geometry program for 
three- and four-dimensional Minkowski space. That is, characterize three- and 
four-dimensional Minkowski space in terms of a group of motions (^,0). Second, we 
must determine what additional structure on the ordered set / would yield from 
Tomita-Takesaki theory algebraic relations among the J,- (and hence, among the x,) 
which coincide with the algebraic characterization found in step one. 

The organization of the thesis is as follows: in Chapter 2, the given pair (G,<3) 
is used to construct a three-dimensional Minkowski space out of the plane at infinity. 
Then identification of the involutory elements of Q with spacelike lines and their 
group action in with reflections about spacelike lines is made. 

In Chapter 3 using the same initial data, (^,0), as was given in Chapter 2 but 
satisfying different axioms, a four-dimensional Minkowski space is constructed. The 
approach taken here differs from that taken in Chapter 2. This time the affine space 
is constructed first and then the hyperplane at infinity is used to obtain the metric. 
The identification of the elements of Q with spacelike planes and their group action 
in with reflections about spacelike planes is made. 

In Chapter 4 a concrete example of the three-dimensional characterization is 
given. As already mentioned, Bisognano and Wichmann showed that for quantum 
field theories satisfying the Wightman axioms the modular objects associated by 



6 

Tomita-Takesaki theory to the vacuum state and local algebras in wedgelike regions 
in three-dimensional Minkowski space have geometrical interpretation [5]. In 
particular, the modular conjugations, {J/},e/, act as reflections about spacelike lines. 
In this chapter it is shown that if one chooses the set of wedgelike regions as the 
index set /, the group JT" generated by the set {J/}/e/ satisfies the axiom system given 
in Chapter 2 for the construction of a three-dimensional Minkowski space. 

In Chapter 5 some concluding remarks about the second step described above 
are made. It is noted that if one assumes the modular stability condition [6] and the 
half-sided modular inclusion relations given by Wiesbrock [29], then one does obtain 
a unitary representation of the 2+1-dimensional Poincare group. 






4 



CHAPTER 2 
A CONSTRUCTION OF THREE-DIMENSIONAL MINKOWSKI SPACE 



In this chapter we give an absolute geometric, that is, an algebraic, 
characterization of three-dimensional Minkowski space. This chapter is a version of a 
preprint by the author entitled "A Group-Theoretic Construction Of Minkowski 
3-Space Out Of The Plane At Infinity" [28]. Along with the well-known 
mathematical motivations [1, 2] there are also physical motivations, as we discussed 
in Chapter 1. Three-dimensional Minkowski space is an affine space whose plane at 
infinity is a hyperbolic projective-metric plane [12]. In "Absolute Geometry" [2], 
Bachmann, Pejas, Wolff, and Bauer (BPWB) took an abstract group generated 
by an invariant system Q of generators in which each of the generators was 
involutory, satisfying a set of axioms and constructed a hyperbolic projective-metric 
plane in which the given group (5 was isomorphic to a subgroup of the group of 
congruent transformations (motions) of the projective-metric plane. By interpreting 
the elements of Q as line reflections in a hyperbolic plane, BPWB showed that the 
hyperbolic projective-metric plane could be generated by these line reflections in 
such a way that these line reflections form a subgroup of the motions group of the 
projective-metric plane. 

Coxeter showed in [13[ that every motion of the hyperbolic plane is generated 
by a suitable product of orthogonal line reflections, where an orthogonal line 
reflection is defined as a harmonic homology with center exterior point and axis the 
given ordinary line and where the center and axis are a pole-polar pair. Here we 
show that Coxeter's and BPWB's notions of motions coincide in the hyperbolic 



8 
projective-metric plane and that the motions can be viewed as reflections about 

exterior points. 

Next we embed our projective-metric plane into a three-dimensional projective 
space. By singling out our original plane as the plane at infinity, we obtain an affine 
space whose plane at infinity is a hyperbolic projective-metric plane, 
three-dimensional Minkowski space. Finally, we show that the motions of our 
original plane induce motions in the affine space and, by a suitable identification, we 
show that any motion in Minkowski space can be generated by reflections about 
spacelike lines. Thus, to construct a three-dimensional Minkowski space, one can 
start with a generating set Q of reflections about spacelike lines in the plane at 
infinity. So Q may be viewed as a set of reflections about exterior points in a 
hyperbolic projective-metric plane. Out of the plane at infinity, one can obtain a 
three-dimensional affine space with the Minkowski metric, which is constructed from 
a group generated by a set of even isometries or rotations. 

The approach in this chapter differs from the method used by Wolff [30] for 
two-dimensional Minkowski space and by Klotzek and Ottenburg [19] for 
four-dimensional Minkowski space. The approach in these papers is to begin by 
constructing the affine space first. For Wolffs [30] two-dimensional case, the 
elements of the generating set Q are identified with line reflections in an affine plane. 
For Klotzek and Ottenburg's [19] four-dimensional case, the elements of the 
generating set Q are identified with reflections about hyperplanes in an affine space. 
Thus, in each of these papers, the generating set G is identified with a set of 
symmetries or odd isometries. A map of affine subspaces is then obtained using the 
definition of orthogonality given by commuting generators. This map induces a 
hyperbolic polarity in the hyperplane at infinity, yielding the Minkowski metric. 

To briefly recap the two approaches described above, note that both 
approaches, ours and the one given by Klotzek and Ottenburg [19] and by Wolff 



9 
[30], start with a generating set Q of involution elements. In our approach, one can 
identify the elements of Q with a set of even isometries (rotations) and use the 
definition of orthogonality induced by the commutation relations of the generators in 
the hyperplane at infinity to obtain the polarity and then embed this in an affine 
space to get Minkowski space. In the approach of Klotzek and Ottenburg [19] and 
Wolff [30], one can identify the elements of Q with a set of odd isometries 
(symmetries), construct an affine space first, and then use the definition of 
orthogonality induced by the commutation relations of the generators in the affine 
space to obtain a polarity in the hyperplane at infinity to get Minkowski space. 

2.1 Preliminaries 

The starting point for an algebraic characterization of Minkowski space is 
therefore far from unique. Our particular choice of algebraic characterization, in 
terms of reflections about spacelike lines in three dimensional Minkowski space, is 
motivated by physical considerations [6] which we briefly explain in the conclusion. 

A hyperbolic projective-metric plane is a projective plane in which a 
hyperbolic polarity is singled out and used to define orthogonality in the plane. A 
polarity is an involutory projective correlation. A correlation is a one-to-one 
mapping of the set of points of the projective plane onto the set of lines, and of the 
set of lines onto the set of points such that incidence is preserved. A projective 
correlation is a correlation that transforms the points 7 on a line b into the lines y 
through the corresponding point B' . So, in general, a correlation maps each point A 
of the plane into a line a of the plane and maps this line into a new point A' . When 
the correlation is involutory, A' always coincides with A. Thus a polarity relates A to 
a, and vice versa. A is called the pole of a and a is called the polar of ^. Since this is 
a projective correlation, the polars of all the points on a form a projectively related 
pencil of lines through A. 



10 

The polarity dualizes incidences: if A lies on b, then the polar of A, a, contains 
the pole of b,B. In this case we say that A and B are conjugate points, and that a 
and b are conjugate lines, li A and a are incident, then A and a are said to be 
self-conjugate: A on its own polar and a through its own pole [14]. A hyperbolic 
polarity is a polarity which admits self-conjugate points and self-conjugate lines. The 
set of all self-conjugate points is called a conic, which we shall call the absolute. 

In a projective plane in which the theorem of Pappas and the axiom of Fano 
hold, the polarity can be used to introduce a metric into the plane. Orthogonality is 
defined as follows: two lines (or two points or a line and a point) are said to be 
orthogonal or perpendicular to each other if they are conjugate with respect to the 
polarity. 

Congruent transformations of the plane are those coUineations of the plane 
which preserve the absolute; that is, those coUineations which leave the absolute 
invariant. In a projective plane with a hyperbolic polarity as absolute, the group of 
all coUineations in the plane leaving the absolute invariant is called the hyperbolic 
metric group and the corresponding geometry is called the hyperbolic metric 
geometry in the plane [31]. 

The conic or absolute, separates the points of the projective plane into three 
disjoint classes: ordinary or interior points, points on the absolute, and exterior 
points. The lines of the projective plane are likewise separated into three disjoint 
classes. Secant lines, lines that contain interior points, exterior points, and precisely 
two points on the absolute. Exterior lines, lines that contain only exterior points. 
And tangent lines, lines that meet the absolute in precisely one point and in which 
every other point is an exterior point. 

Definition 2.1.1. Two lines containing ordinary points, two secant lines, are said to 
be parallel if they have a point of the absolute in common. 
Remark. The set of all interior points and the set of lines formed by intersecting 



11 

secant lines with the set of interior points, ordinary lines, is classical hyperbolic 
plane geometry. 

2.2 Construction of IT 

In this section we list the axioms and main results of BPWB [2] and provide a 
sketch of some of the arguments they used which are pertinent to this work. For 
detailed proofs, one is referred to the work of BPWB [2]. 

Definition 2.2.1. A set of elements of a group is said to be an invariant system if it 
is mapped into itself (and thus onto itself) by every conjugation by an element of the 
group. An element a of a group is called an involution if a- = 1©, where 1© is the 
identity element of the group 0. ■ -' •• - « 

Basic assumption: A given group is generated by an invariant system Q of 
involution elements. 

The elements of Q are denoted by lowercase Latin letters. Those involutory 
elements of that can be represented as ab, where a,b e Q, are denoted by 
uppercase Latin letters. If ^,t| e and ^r| is an involution, we denote this by 4|r|. 
Axioms 

Axiom 1: For every P and Q there is a g with P, Q \g. 
Axiom 2: IfP,Q \g,h then P = Q or g = h. 
Axiom 3: Ifa,b,c \P then abc = d e Q. 
Axiom 4: Ifa,b,c \g then abc = d e Q. 
Axiom 5: There exist g,h,j such that g \h butj / g,h,gh. 

Axiom 6.- There exist elements d,a,b e Q such that d,a,b / P,c for P,c e 0. (There 
exist lines which have neither a line nor a point in common.) 

Axiom 7: For each P and for each g there exist at most two elements hj e Q such 
that P\h,j butg,h / A,c and gj / B,d for anyA,B,c,d e <S>(that is, have neither a 
point nor a line in common). 



12 
Axiom 8: One never has P = g- 
We call the set of axioms just given axiom system A, denoted by as- A. 

The initial interpretation of the elements of Q is as secant or ordinary lines in 
a hyperbolic plane for BPWB [2]. In our approach, we view the elements of Q 
initially as exterior points in a hyperbolic plane. After embedding our hyperbolic 
projective-metric plane into an affme space, we can identify the elements of Q, our 
generating set, with spacelike lines and their corresponding reflections in a 
three-dimensional Minkowski space. By realizing that statements about the geometry 
of the plane at infinity correspond to statements about the geometry of the whole 
space where all lines and all planes are considered through a point, we see that the 
axioms also are statements about spacelike lines, the elements of Q, and timelike 
lines, the elements P of 0, through any point in three-dimensional Minkowski space. 

The models of the system of axioms are called groups of motions; that is, a 
group of motions is a pair (0, Q) consisting of a group and a system Q of 
generators of the group (5 satisfying the basic assumption and the axioms. 

To give a precise form to the geometric language used here to describe 
group-theoretic concepts occurring in the system of axioms, we associate with the 
group of motions (JS>,<3) the group plane (!S>,Q), described as follows. 

The elements of Q are called lines of the group plane, and those involutory 
group elements that can be represented as the product of two elements of Q are 
called points of the group plane. Two lines g and h of the group plane are said to be 
perpendicular if g \h. Thus, the points are those elements of the group that can be 
written as the product of two perpendicular lines. A point P is incident with a line g 
in the group plane if P \g. Two lines are said to be parallel if they satisfy Axiom 6. 
Thus, \i P ^ Q, then by Axioml and Axiom 2, the points P and Q in the group 
plane are joined by a unique line, li P )( g then Axiom 7 says that there are at most 
two lines through P parallel to g. 



13 
Lemma 2.2.2.[2| For each a e 0, the mappings Ga '■ g ^ g"^ = CLga and 
Ga '■ P >—>■ P^ = clPol are one-to-one mappings of the set of lines and the set of points, 
each onto itself in the group plane. 

Proof: Let a e 0, and consider the mapping y h- >■ y« s aya of onto itself. It is 
easily seen that this mapping is bijective. Because Q is an invariant system (a* e Q 
for every a,b e ^) ^ is mapped onto itself, and if P is a point, so that P - gh with 
g\h, then P" = g"/?" and g«|/7«, so that P« is also a point. Thus, g ^ g^, p ^ p» 
are one-to-one mappings of the set of lines and the set of points, each onto itself in 
the group plane. ■ 

Definition 2.2.3. A one-to-one mapping a of the set of points and the set of lines 
each onto itself is called an orthogonal collineation if it preserves incidence and 
orthogonality. 

Since the " | " relation is preserved under the above mappings, the above 
mappings preserve incidence and orthogonality as defined above. 
Corollary 2. 2. 4. [2] The mappings 

Cf a '■ g ^-^ g'^ ^nd Ua '■ P '-^ P'^ 
are orthogonal collineations of the group plane and are called motions of the group 
plane induced by a. 

In particular, if a is a line a we have a reflection about the line a in the group 
plane, and if a is a point A, we have a point reflection about A in the group plane. 

If to every a e one assigns the motion of the group plane induced by a, 
one obtains a homomorphism of onto the group of motions of the group plane. 
Bachmann |1] showed that this homomorphism is in fact an isomorphism so that 
points and lines in the group plane may be identified with their respective 
reflections. Thus, is seen to be the group of orthogonal collineations of 
generated by Q. 



14 
Definition 2.2.5. Planes that are representable as an isomorphic image, with respect 
to incidence and orthogonaUty, of the group plane of a group of motions (0, Q), are 
called metric planes. 

BPWB showed how one can embed a metric plane into a projective-metric 
plane by constructing an ideal plane using pencils of lines [2]. We shall now outline 
how this is done. _. ,^ 

Definition 2.2.6. Three lines axe said to lie in a pencil \i their product is a line; that 
is, a,b,c lie in a pencil if ■' / - ► 

abc = d e Q. (♦) 

Definition 2.2.7. Given two lines a,b with a ^ b, the set of hues satisfying (*) is 
called a pencil of lines and is denoted by G(ab), since it depends only on the product 
ab. 

Note that the relation (*) is symmetric, that is, it is independent of the order 
in which the three lines are taken, since cba = {abc)~^ is a line, the invariance of Q 
implies that cab = {abcY is a line and that every motion of the group plane takes 
triples of lines lying in a pencil into triples in a pencil. The invariance of Q also 
shows that (*) holds whenever at least two of the three lines coincide. 

Using the given axioms, BPWB [2] showed that there are three distinct classes 
of pencils. U a,b\V then G{ab) = {c : c \V]. In this case, G(ab) is called a pencil of 
lines with center V and is denoted by G(V). If a,b \c then G(ab) = {d : d \c}. In this 
case, G(ab) is called a pencil of lines with axis c and is denoted by G(c). 

By Axiom 6, there exist lines a, b, c which do not have a common point or a 
common line. Recall that lines of this type are called parallel. Thus, in this case 
G(ab) = {c : c \\ a,b where a \\ b}, which we denote by Px. 

Using the above definitions of pencils of lines and the above theorems, BPWP 
[2] proved that an ideal projective plane, n, is constructed in the following way. An 



15 
ideal point is any pencil of lines G(ab) of the metric plane. The pencils G(P) 
correspond in a one-to-one way to the points of the metric plane. An ideal line is a 
certain set of ideal points. There are three types: 

1. A proper ideal line gia), is the set of ideal points that have in common a line a 
of the metric plane. 

2. The set of pencils G{x) with x \P for a fixed point P of the metric plane, which 
we denote by P. 

3. Each set of ideal points that can be transformed by a halfrotation about a fixed 
point P of the metric plane into a proper ideal line, which we denote by /?». 

The polarity is defined by the mappings 

GiQ ^ C and C ^ G(0; 
Px I— ♦ Poo and pao I— ► Px', 
Gic) 1— ► gic) and gic) h-» G(c). 

Bachmann [1] showed that the resulting ideal plane is a hyperbolic projective plane 
in which the theorem of Pappus and the Fano axiom hold; that is, it is a hyperbolic 
projective-metric plane. 

In this model, the ideal points of the form G{P) are the interior points of the 
hyperbolic projective-metric plane. Thus the points of the metric plane correspond in 
a one-to-one way with the interior points of the hyperbolic projective-metric plane. 
The ideal points G{x), for jc e ^ are the exterior points of the hyperbolic 
projective-metric plane. v - . 

Theorem 2.2.8. Each x e Q corresponds in a one-to-one way with the exterior points 
of the hyperbolic projective-metric plane. - - 

Proof: Because each line d of the metric plane is incident with at least three points 
and a point is of the form ab with a \ b, then each x 6 ^ is the axis of a pencil. From 
the uniqueness of perpendiculars each x e Q corresponds in a one-to-one way with 
the pencils G{x). Hence, each x e Q corresponds in a one-to-one way with the 
exterior points of the hyperbolic projective-metric plane. ■ 



16 

Thus, the axioms can be viewed as axioms concerning the interior and exterior 
points of a hyperbolic projective-metric plane. The ideal points of the form G(ab) 
where a \\ b are the points on the absolute, that is, the points at infinity in the 
hyperbolic projective-metric plane. 

Now consider the ideal lines. A proper ideal line g{d) is a set of ideal points 
that have in common a line a of the metric plane. 
Theorem 2.2.9. A proper ideal line g(a) is a secant line of the form 

g{a) = {P,x,G{bc) : x,P\a and abc e Q where b \\ c). 

Proof: Every two pencils of lines of the metric plane has at most one line in 
common. By Axiom 7, each line belongs to at most two pencils of parallels and each 
line g e Q belongs to precisely two such pencils. Thus, a proper ideal line contains 
two points on the absolute, interior points, and exterior points; that is, a proper 
ideal line is a secant line. If we identify the points P with the pencils G{P) and the 
lines X with the pencils G(x), then a secant line is the set g{c) - {P,x,G{ab) : x,P \c 
and abc e G where a \\ b}. ■ 

Corollary 2.2.10. The ideal line which consists of pencils G{x) with x \P for a fixed 
point P of the metric plane consists of only exterior points; that is, it is an exterior 
line. Under the identification ofx with G(x) then P = {x e Q : x\P}. 

The last type of ideal line is a tangent line. It contains only one point, 
G(ab) = Poo, on the absolute. Denoting this line by poo, then 
Poo = {Giab)} u {x e g : abx e Q}, where a || b. Recalling that each x e g 
corresponds to an exterior point in the hyperbolic projective-metric plane, we see 
that a tangent line consists of one point on the absolute and every other point is an 
exterior point. 

Also note that under the above identifications, each secant line g{c) 
corresponds to a unique "exterior point", c, c € g{c) since one only considers those 



17 
x,P \c such that jcc 9^ 1© and Pc ^ 1©. Each exterior Une corresponds to a unique 
interior point P and each tangent line corresponds to a unique point on the absolute. 
Theorem 2.2.11. The map O given by 
(i) 0(c) = g(c), 0(g(c)) =c 
(ii) (D(P) = p, 0(p) = P 

(Hi) O(poo) = Poo, <I>(Pao) = Poo 

75 a polarity. 

Proof: Let V be the set of all points of H and C the set of all lines of IT. From the 

remarks above it follows that O is a well-defined one-to-one point-to-line mapping of 

V onto C and a well-defined one-to-one line-to-point mapping of C onto V. Next we 

show that O is a correlation and for this it suffices to show that C) preserves 

incidence. 

Let g{c) = {P,x,G{ab) : x,P\c and abc e Q where a || 6} be a secant line. Let 
A,B,d,Pa, € g{c), where Poo = G(eJ) = {x e G : xef e G and e ||/}.Then A,B,d\c and 
cab e Q. ^:'. . '' , •?■ t ^■'h 

^{A) = A = {x:x\A}, ^(B) = B, 0(d) = gid), 0(Poo) = p^ = {Poo} <J {x : ejk e Q} 
cD(g(c)) =c eAnBngid)np^ and a>(g(c)) e 0(^), 0(5), 0(J), <I)(Poo). 

Hence, O preserves incidence on a secant line. Now consider an exterior line 

P = {x : x\P} and let a,b e P. Then a,b\P and it follows that P e gia)ngiby, that 

is, 0(P) e 0(a) n 0(6) and O preserves incidence on an exterior line. 

Finally, let /7oo = {G(ab)} u {x : abx e Q where o || b} be a tangent line. 
Clearly, since OiGiab)) = p^o, then Poo = G(aZj) e /?oo. Now suppose that d € /?oo. 
Then a^^/ e Q and 

^(^ = M^ = {A,x,G(ef) : ^,jc|i/ and Je/ e G where e || /}. - 

Thus, d 6 G(o6) n G(ey) and Poo 6 gid). This implies that d e poo and 0(/7oo) e ^(d). 
Hence, O preserves incidence and is a correlation. 



18 
Note also that from the work above, O transforms the points 7 on a hne b into 
the Unes 0(Y) through the point 'I>(b). Thus, <1) is a projective correlation. Since 
O^ = 1©, then O is a polarity. Moreover, since 0(pc») - Paa with Poo e /?<», then O is 
a hyperbolic polarity. ■ 

Theorem 2.2.12 The definition of orthogonality given by the polarity agrees with 
and is induced by the definition of orthogonality in the group plane. 
Proof: If we define perpendicularity with respect to our polarity then the following 
are true (we use the notation •O' to denote the phrase "if and only if "): 
(i) g{c) 1 g{d) <^ 0(^c)) = c e gia) and Oigia)) ^ a e gic) <^ a \c. 

(ii) gic) IP <r^ (^(g(c)) = c e P <-> c IP. 

(iii) CIP <^ cD(P) = P e gic) = <D(c) <^ P \c. 

(iv) C Igix)^^ ^(gix)) =x e g(c) = cD(c) <^ x \c. ■ 

Instead of interpreting our original generators as ordinary lines in a hyperbolic 
plane, we now interpret them as exterior points. We can construct a hyperbolic 
projective-metric plane in which the theorem of Pappus and Fano's axiom hold, 
which is generated by the exterior points of the hyperbolic projective-metric plane. 

With the identifications above and the geometric objects above, we show in 
the next section that the motions of the hyperbolic projective-metric plane above can 
be generated by reflections about exterior points; that is, any transformation in the 
hyperbolic plane which leaves the absolute invariant can be generated by a suitable 
product of reflections about exterior points. 

2.3 Reflections About Exterior Points 
Definition 2.3.1. A collineation is a one-to-one map of the set of points onto the set 
of points and a one-to-one map of the set of lines onto the set of lines that preserves 
the incidence relation. 
Definition 2.3.2. A perspective collineation is a collineation which leaves a line 



19 
pointwise fixed, its axis, and a point line-wise fixed, its center. 

Definition 2.3.3. A homology \s a perspective collineation with center a point B and 
axis a line b where B is not incident with b. 

Definition 2.3.4. A harmonic homology with, center B and axis b, where B is not 
incident with 6, is a homology which relates each point A in the plane to its 
harmonic conjugate with respect to the two points B and {b,[A,B\), where [A,B\ is 
the line joining A and B and {b,[A,B\) is the point of intersection of b and [^,5]. 
Definition 2.3.5. A complete quadrangle is a figure consisting of four points (the 
vertices), no three of which are collinear, and of the six lines joining pairs of these 
points. If / is one of these lines, called a side, then it lies on two of the vertices, and 
the line joining the other two vertices is called the opposite side to /. The 
intersection of two opposite sides is called a diagonal point. 

Definition 2.3.6. A point D is the harmonic conjugate of a point C with respect to 
points A and B il A and B are two vertices of a complete quadrangle, C is the 
diagonal point on the line joining A and B, and D is the point where the line joining 
the other two diagonal points cuts [^,5]. One denotes this relationship by 
HiAB,CD). 

Example 2.3.7. Let A,B, and C be three collinear points. For a quick construction of 
the harmonic conjugate D of C with respect to A and B let Q,R,S be any points such 
that [2,/?],[!5,5], and [R,S] pass through ^4, 5, C respectively. Let 
{P} = [A,S] n [B,R], then {D} = [A,B] n [P,Q] ([11]). Note that if [R,S\ \\ [A,Q then 
D is the midpoint of A and B. 

Coxeter [13] showed that any congruent transformation of the hyperbolic 
plane is a collineation which preserves the absolute and that any such 
transformation is a product of reflections about ordinary lines in the hyperbolic 
plane where a line reflection about a line w is a harmonic homology with center M 
and axis m, where Mand m are a pole-polar pair and M is an exterior point. A point 



20 . ■ ' - ■ 

reflection is defined similarly, a harmonic homology with center M and axis m, where 
M and m are a pole-polar pair, M is an interior point, and m is an exterior line. Note 
that in both cases, M and /w are nonincident. 

In keeping with the notation employed at the end of §2. 2, let b be an exterior 
point and g{b) its pole. 



r 

A 



Lemma 2.3.8. The map T^ : < 



Ab 


and 


d 


1 — ► 


db 


-\ 


A' 




d 


1 — > 


gid)' 




Pi 




Poo 


1 — ► 


Poo 


y 



A ^ A" d ^ ^d)" y JS. 

Poo H 



coUineation. 

Proof: This follows from the earlier observation that the motions of the group plane 

map pencils onto pencils preserving the "|" relation. ■ 

Lemma 2.3.9. ^/, is a perspective coUineation and, lience, a homology. 

Proof: Recall that g(6) = {A,x,Pca : x,A \b and where b lies in the pencil Pa,}. For ' 

any A and :>i: in g(b) we have A^ = A and x* = jc since A,x \b and ii A' ,x' g g(b) then 

A',x' / b and A^ ^ A, x^ * x, and ^*,x* / b. Thus, A'',x^ € gib). 

Recall also that Poo = G{cd), where c and d do not have a common 
perpendicular nor a common point and thus, G{cd) = {f : fed e Q). Now g{b) is a 
secant line, so that it contains two such distinct points, /•<» and goo, say, on the 
absolute. Since the motions of the group plane map pencils onto pencils preserving 
the "I" relation it follows that if c,^ e Poo then c*, J* e Poo and hence, P^ = Poo and 

Qi = Qoo. Moreover, if ;?oo g g{b) then it follows that R^ ^ gib). Thus, 4^/, leaves 

""' ''■ ■' ?■> . 
gib) pomtwise mvariant. •&■► v . j'*,/ ,, ^ 

Now let g{d), Q, and Too be a secant line, exterior line, and tangent line, 

respectively, containing b. For e e gid) we have e \d and e^ \ d^ = d since b | d, 

thus e* e gid). For A e gid), A^ \d^ = d, so A^ e gid). Similarly, it follows that if 

Poo e gid) then P* e gid), and that gid)'' = gid). One easily sees that Q'' = Q and 

ri = /-CO. Thus, ^^h leaves every line through b invariant and ^b is a perspective 



21 
collineation for each b e Q. U 
Theorem 2.3.10. ^i, 's a harmonic homology. 

Proof: Since A^ is again a point in the original group plane and since d'' is again a 
line in the original group plane and from the observations above, we have, for each 
b e G, ^f, maps interior points to interior points, exterior points to exterior points, 
points on the absolute to points on the absolute, secant lines to secant lines, exterior 
lines to exterior lines, and tangent lines to tangent lines. Moreover, since C^*)* = t, 
for any ^ e 0, 4^^ is involutory for each b e Q. Now in a projective plane in which 
the theorem of Pappus holds, the only coUineations which are involutory are 
harmonic homologies [10], thus ^h is a harmonic homology for each b e Q. M 
Theorem 2.3.11. Interior point rejections are generated by exterior point re/lections. 
Proof: A similar argument shows that for each interior point A, 4^^ is a harmonic 
homology with center A and axis A where A is the polar oi A, A € A, and where 4^^ 
is defined analogously to 4'/,. Thus, each T^ is a point reflection and since A is the 
product of two exterior points, we see that point reflections about interior points are 
generated by reflections about exterior points. ■ 

Theorem 2.3.12. The rejection of an interior point about a secant line is the same 
as reflecting the interior point about an exterior point. Moreover, since any motion 
of the hyperbolic plane is a product of line reflections about secant lines, any motion 
of the hyperbolic plane is generated by reflections about exterior points. 
Proof: Consider a line reflection in the hyperbolic plane; that is, the harmonic 
homology with axis g{b) and center b. Let A be an interior point and g{d) a line 
through A meeting b. Since b e g(d) then b \d and g{d) is orthogonal to g(b). Let E 
be the point where g(b) meets g(d). Since E e gib) then E \b and Eb =/for some 
/ e ^. It follows that the reflection of A about g(b) is the same as the reflection of A 
about E. Since b \d and E\d then bd = C and we have E,C \b,d with b ^ d. Thus, by 
Axiom 2, E = C = bd. Hence, A'' = A""" = AK ■ 



22 
Theorem 2.3. 13 Reflections of exterior points about exterior points and about 
exterior lines are also motions of the projective-metric plane; that is, the ^b 's for 
b e Q acting on exterior points and exterior lines are motions of the hyperbolic 
projective-metric plane. 

Proof: The motions of the projective-metric plane are precisely those collineations 
which leave the absolute invariant. ■ 

We also point out that the proof that each ^F^ is an involutory homology also 
showed that the Fano axiom holds, since in a projective plane in which the Fano 
axiom does not hold no homology can be an involution [4]. 

2.4 Embedding a Hyperbolic Projective- Metric Plane 
In this section we embed our hyperbolic projective-metric plane into a three 
dimensional projective space, finally obtaining an affine space whose plane at infinity 
is isomorphic to our original projective-metric plane. Any projective plane Ft in 
which the theorem of Pappus holds can be represented as the projective coordinate 
plane over a field /C. (The theorem of Pappus guarantees the commutivity of /C.) 
Then by means of considering quadruples of elements of /C, one can define a 
projective space P3(/C) in which the coordinate plane corresponding to Tl is included. 
If the Fano axiom holds, then the corresponding coordinate field JC is not of 
characteristic 2 [4]. By singling out the coordinate plane corresponding to 11 as the 
plane at infinity, one obtains an affine space whose plane at infinity is a hyperbolic 
projective-metric plane: that is, three-dimensional Minkowski space. 

To say that a plane n is a projective coordinate plane over a field /C means 
that each point of n is a triple of numbers (xo,xi,X2), not all x, = 0, together with 
all multiples (kxo,hc\,'kx2), for X ;t and X e /C. Similarly, each line of 11 is a triple 
of numbers [«o,"i,"2], not all w, = 0, together with all multiples [Xuo,Xu\,Xu2], 
A, 9t 0. In PsC/C), all the quadruples of numbers with the last entry zero correspond 



fr^^-F-^.g— ,»r»-.' -' 



23 
to n. One can now obtain an affine space A by defining the points of A to be those 
of P3(/C) - n; that is, those points whose last entry is nonzero; a Une / of ^ to be a 
Une /' in P3(/C) - n minus the intersection point of the Une /' with FI; and by 
defining a point P in A to be incident with a Une I oi A if, and only if, P is incident 
with the corresponding /'. Planes of A are obtained in a similar way [14]. 

Thus, each point in n represents the set of all lines in A parallel to a given 
line, where lines and planes are said to be parallel if their first three coordinates are 
the same, and each line in FI represents the set of all planes parallel to a given plane. 
Because parallel objects can be considered to intersect at infinity, we call n the 
plane at infinity. 

2.5 Exterior Point Reflections Generate Motions in an Affine Space 
In this section we state and prove the main result of this chapter. Coxeter 
showed that three-dimensional Minkowski space is an affine space whose plane at 
infinity is a hyperbolic projective- metric plane [11]. He also classified the lines and 
planes of the affine space according to their sections by the plane at infinity as 
follows: 

Line or Plane Section at Infinity 

Timelike line Interior point 

Lightlike line Point on the absolute 

Spacelike line Exterior point 
Characteristic plane Tangent line 

Minkowski plane Secant line 

Spacelike plane Exterior line 

He also showed that if one starts with an affine space and introduces a 
hyperbolic polarity in the plane at infinity of the affine space, then the polarity 
induces a Minkowskian metric on the whole space. Under a hyperbolic polarity, a 
line and a plane or a plane and a plane are perpendicular if their elements at infinity 



24 
correspond. Two lines are said to be perpendicular if they intersect and their 
elements at infinity correspond under the polarity. 

Theorem 2.5.1. Every motion in a three-dimensional affme space with a hyperbolic 
polarity deRned on its plane at infinity, is generated by reflections about exterior 
points. Moreover, because exterior points correspond to spacelike lines, then any 
motion in three-dimensional Minkowski space is generated by reflections about 
spacelike lines. •if ,.■ . i. t 

Proof: Because any motion in three-dimensional Minkowski space can be generated 
by a suitable product of plane reflections, it suffices to show that reflections about 
exterior points generate plane reflections. 

Let a be any Minkowski plane or spacelike plane. Let P be any point in 
Minkowski space. Let / be the line through P parallel to a. Let ttoo denote the 
section of a at infinity. Applying the polarity to aoo we get a point goo -L aoo. Let g 
be a line through P whose section at infinity is g-oo, so that g is a line through P 
orthogonal to a. because each line in the plane at infinity contains at least 3 points, 
there exists a line / in a which is orthogonal to g as g=o -L aoo. Now let m be a line 
through P not in a which intersects /. It follows that the reflection of P about a is 
the same as reflecting m about / and taking the intersection of the image of m under 
the reflection with g. By the construction of the affme space and the definition of 
orthogonality in the affine space it follows that / and m must act as their sections at 
infinity act. because any point reflection in the hyperbolic projective-metric plane 
can be generated by reflections about exterior points, we have that the reflection of 
P about a is generated by reflections of P about spacelike lines. ■ 

2.6 Conclusion 
As already indicated above, the geometric model for the generators of Q which 
lies behind the choice of algebraic characterization of three-dimensional Minkowski 



25 
space differs significantly from those of previous absolute geometric characterizations 
of Minkowski space. The model given here is the set of reflections about spacelike 
lines, which is not a choice which would be made a priori by other mathematicians. 
However, this is yet another example of a situation where the initial data are 
imposed by physical, as opposed to purely mathematical, considerations. 

In the next chapter starting with the same initial data, but satisfying different 
axioms, a construction of four-dimensional Minkowski space is given. First the affme 
space is constructed and then the hyperplane at infinity is used. Also given is an 
explicit construction of the field, the vector space, and the metric. 









\ , ,• .,.'.•< ,' wiv^v 






CHAPTER 3 
A CONSTRUCTION OF FOUR-DIMENSIONAL MINKOWSKI SPACE 



In this chapter a construction of four-dimensional Minkowski space will be 
given using the same initial data as in Chapter 2 but satisfying different axioms. The 
actual construction is quite long, so to aid the reader in following, A brief outline of 
the procedure shall be given here. First some general theorems and the basic 
definitions will be given in the first two sections. In Sections 3.3 and 3.4 attention is 
restricted to two dimensions in order to obtain the necessary machinery to construct 
the field. Once the field has been obtained, then a vector space is constructed and 
given a definition of orthogonality. It is then shown that the vector definition of 
orthogonality is induced by and agrees with the initial definition of perpendicularity 
for the geometry generated by the original set of involutions Q. 

To obtain a metric vector space from the constructed vector space V, a map 7i 
is defined on the subspaces of V. The map n is defined to send a subspace to its 
orthogonal complement. Using the work of [3] (which is given for the convenience of 
the reader) the Minkowski metric is obtained and hence, Minkowski space. The last 
section of this chapter identifies the elements of G with spacelike planes in 
Minkowski space and the motions of the elements of Q with reflections about 
spacelike planes, as was required in Chapter 1. 

3.1 Preliminaries and General Theorems 
Let there be given a nonempty set G of involution elements and the group (d it 
generates, where for any a e G and for any ^ e we have a^ = ^"'a^ e G- If the 
product of two distinct elements ^1,^2 e is an involution then we denote this by 

26 



27 
writing ^i |^2- We note that if ^i |£,2 then qj\^2 fo^ ^^^ ^^^xAl iii ® because . 

^t^i = ^-i^i^r'^2^ = r'^1^2^ = r'^2^i4 = ^ki- , 

Let M = {ap: a | P and a,p e ^} and V = Q vj M. We consider the 
elements of Q as spacelike planes and the elements of M. as Minkowskian or 
Lorentzian planes. Thus, V consists of the totality of "non-singular" planes and we 
denote the elements of "P by a, p,y, 

We begin by giving the basic assumption and by making some preliminary 
definitions. All the axioms are then listed for the convenience of the reader. The 
geometric meaning of the axioms and the symbols used will be made clear in the 
appropriate sections. The first four sections examine the incidence axioms. The order 
axioms are reintroduced in Section 3.5, where we construct and order the field to 
obtain a field isomorphic to the reals. In Section 3.6 we give the motivation behind 
the particular choice of dilation axioms. Using these axioms we are able to define a 
scalar multiplication and thereby obtain an affine vector space. The polarity axioms 
are given again and examined in Section 3.8, where we define orthogonal vectors. 
Basic assumption: If a e Q and ^ & M then a^ e Q and ^^ e M for every E, in Q. 

For the following, let a, P e "P with a | p. 
Definition 3.1. If a,p e Q, then aP 6 A^ by definition and we write alp and we 
say a is perpendicular to or orthogonal to p. ' - 

Definition 3.2. Suppose that a e ^, P e M.. ' " ■ ^ 

(i) If aP e Q, then we write a -L P and we say a is perpendicular top. 
(ii) If aP g Q, then we write al P and we say a is absolutely perpendicular to p. 
Definition 3.3. Let X = {aP : alp}. We call the elements of X points and we 

denote these elements hy A, B,C, 

Definition 3.4. If a,p e M and aP e M, then we write a 1 P and we say a is 
perpendicular to p. 



•'I.- 



28 
Definition 3.5. The point A and the plane a are called inddent when A\a. For each 
a e V, set Xa = {A : ^ |a}.se that ^,5|ai,a2 where A ^ B and ai 5^ a2. 

Suppose that A,B\a.\,a.2; where A ^ B and ai ^ ai- We define the line g 
containing A and B as 

g = gAB = [ct],ci2] = {C e X : C|ai,a2}. 

We say that g is the intersection of ai and aa (-^a, and Xaj)^ g <^ oci,a2 
{g a Xa\,Xa2)- The Y>omt C \s incident with, g, C & g, ifC|ai,a2. 

li A ^ B are two points such that there exist a and P with A,B\a,^ and a ± P 
then we say that A and B are JoinabJe and we write A,B e gAB - [a, P,ap]. If g- is a 
line which can be put into the form g = [a, P, aP] where a -L P then we say that g is 
nonisotropic. 

If A and B are two distinct points such that there do not exist a, P with 
A,B\fx,^ and a .L P then we say that A and B are unjoinable. If g is a line which 
cannot be put into the form g = [a, p, aP] with a ± P then we say that g is isotropic 
or null. ^-^ ^ ; v' ■ f '■ A 

til y ' '■ ■ 

Incidence Axioms u - ■ ' , ■ ' .- 

Axiom 1. For eacli P,a there exists a unique P e "P such tliat P\^ and aP = Q. 

Axiom 2. 7/^,5 |a,p,s and C\a,^ ttten C\z. 

Axiom 3. IfP,Q\a,^ and a 1 ^ tJien P = Q. 

Axiom 4. //a,p,y e G are distinct and a 1 p 1 y 1 a then apy g g. 

Axiom 5. //a,p e M anda\^ tJien a^ e M. 

Axiom 6. For ail A, B- A ^ B, there exists a,p sucli that A,B\a,^ and a ^ p. 

Axiom 7. If a 1 p then there exists A,B\a.,^ such that A ^ B. 

Axiom 8. ForaJJA,B,C; ABC = D e X. 

Axiom 9. // 0|a,p,y,5 with p,y,5 1 a then py6 = e. 



29 
Axiom 10. If A, B, C are pairwise unjoinable points and A, B \ a then C \ a. 
Axiom 11. For all a e Q, there exist distinct P,y,6 e Q such that a _L P ± y -L ot 

Axiom 12. IfA,B\a; a e G, then there exists P e G such that A,B\^ and ^ 1 a. 
Axiom 13. For each P\ a, a e /A, there are distinct points A, B\a such that 
P ^ A,B and P is unjoinable with both A and B, but A and B are joinable, andifC\u. 
is unjoinable with P then C is unjoinable with A or with B or C = A or C - B. 
Axiom 14. If A and B are joinable and A,B\a., then there exists P ± a such that 

Axiom 15. //a, p,y are distinct with P,y ± a and A,B\a,^,y; A ^ B, then a = Py 
and ifA,B\y; y J. a then a = Py or p = y. 

Order Axioms 

Axiom F. (Formally Real Axiom) [21] Let 0,E \ a, a e M with O and E unjoinable. 

LetX,b,r\ e V. IfO \ X,5,ri andX,b,r\ L a then there is ay e V such that 

0|y, yla, and E^^^^OE^'^^'^ = E^y^^. 

Axiom L. (LUB) If A (z IC, A ^ 0, and A is bounded above, then there exists an A 
in K. such that A > X, for all X in A and ifB > X for all X in A then A < B. 

Dilation Axioms 

Axiom T. IfO e t,g, with t ^ g, where t is timelike or t and g are both isotropic, 

then there exists a unique a e M such that gj (^ Xa- 

Axiom D. (Desargues) Let g,h,k be any three distinct lines, not necessarily coplanar, 

which intersect in a point O. Let P,Q e g; R,S e h; and T,U e k. Ifgpj \\ ggu and 

gRT li gsu then gPR II gQs. 

Axiom R. LetO e g,/T, P,Q e g, and R,S e h. IfgpR \\ ggs then go,POR II go,QOS- 



30 
Polarity Axioms 

Axiom U. (U-^ subspace axiom) Let 0,A,B,T and C be four points with 0,C|y6; 
A,0\a; 0, 5 1 P; with a 1 y and p 1 6. Then there exist X,s e V such that X Iz; 
0,AOB\X; andO,C\^. 

Axiom SI. Ifg cz Xa, CL e G, h c: Xp, ^ e G, and there exist y, 5 e V such that 
Xld; yd e gn> h, g a Xy, and h a X?, then there exists e e G such that g,h c .^s- 
(Ifg and h are two orthogonal spacelike lines then there is a spacelike plane 
containing them.) 

Axiom S2. Let g and h be two distinct lines such that P e gnh but there does not 
exist ^ e V such that g c Xp and h c Xp. Then either there are a,p' e "P such that 
P = ap', g c Xa, and h a Xn' or for all A e g there exists B e h such that P and 
APB are unjoinable. 

This concludes the list of axioms. Note that by Axiom 5, ap i A1 for a _L p, 
a, P e T' and if a, P e M. are distinct with a|P, then.aP e A^. By Axiom 6, every 
two points is contained in a line. 

Notation. Due to the brevity of the theorems and proofs in sections 3.1 
through 3.4, we shall follow the usual convention in absolute geometry [1, 2, 19, 30] 
of simply numbering the results in these sections. 

3.1.1. Properties of M . 

3.1.1.1. The set M * 0. 

Proof: By assumption ^ ^^ 0, so let a e ^. By Axiom 11, there exists y e ^ such 
that a 1 y. Hence, ay e A1 by definition and A1 5^ 0. ■ 

3.1.1.2. The elements of M. are involutions. 

Proof: Let y e A^ . Then we may write y = aP where a, P e ^ and alp. We have 
yy = apap = aapp = 1©. ■ 

3.1.1.3. For every ^ e M. and for every l^ e 0, p^ e M.. 



31 
Proof: Let P = aia2 with ai,a2 e Q and ai |a2. By our assumptions on Q and 
because ai |a2 we have for all ^ e 0, afaj e Q and a, Ittj so that a'^a^ e M. m 

3.1.1.4. The sets Q and M are disjoint, QnM = 0. 

Proof: Let Q' = Q\M. Then Q' consists of involution elements, Q' o M = 0, and 
V=g'^M=Q^M. Let y e ^' and ^ e ©. Now if y" e M. then by 3.1.1.3 above 
y = (y^)^~ G AI. a contradiction. Thus, Q' is an invariant system of generators and 
without loss of generality, we may assume that Q n M = 0. ■ 

3.1.1.5. Ifa e g, ^ e M andal^, then a(3 g M.^ ''^XiX 

Proof: If aP = y e A^ then a = py e ^ where p,y e tM and P|y, which contradicts 
Axiom 5. ■ 

3.1.2. Properties of P. 

For the remainder of this dissertation, let the symbol " <->" denote the phrase 
"if, and only if. 

3.1.2.1. //a,p e V and^ e tJien a 1 P if, and only if, a^ 1 P^. 

Proof: First suppose that a, p e ^. Then a J. p •f^ a^ _L p^ because a ± P implies 
that a| P and a| P <^ a^ | p^ If a e Q and P e AI and a 1 P then aP = y e ^; 
a^,y^ e ^, P^ e M, and a^P^ = y^ e Q implies that a^ 1 p^. Conversely, if 
a^ 1 p^ then a^P^ = 8 e Q and ap = 5^'' e ^, so that a 1 p. Finally, suppose that 
a.p e M. Then a^P^ e M and ap e AI <^ a^P^ e A^. ■ 

3.1.2.2. If a,^ e V and ^ e (3, then a 1^ <^ a'^ 1^^. 

Proof: Let a e C; and p e At and ^ e 0. If a ip then a|p so that a^|p^. Thus, 
a^ 1 p^ or a' 1 p^. If a^ 1 p^ then a^p^ = y e ^. So we have ap = y^"' e g by 
the invariance of ^, which contradicts a ip. Hence, a^ ip^- 

Conversely, suppose that a^lp^. because a^ e g and P^ g Af by the 
invariance of g and M then a^ = y, p^ = 5 imply that y 1 5 and y^"' = ai P = 6^'' 
by the paragraph above. ■ 



32 
3.1.2.3. For each 4 e 0, V^ = V. 
Proof: because V - Q^J M, the result follows from the invariance of Q and M. M 

3.1.3. Properties oi X. 

3.1.3.1. There exists a point; that is, X ^ 0. 

Proof: Let a e ^. By Axiom 11, there exists y, P e ^ such that a, p,y are distinct 
and mutually perpendicular. By Axiom 4, apy - a.b € Q, where 5 = Py e A^ and 
a|5 as a|p,y. Thus, a ±5 and P = a5 is a point. ■ 

3.1.3.2. //0|a,p,y,- a,p,y e a; a 1 p 1 y 1 a; then O = apy. 

Proof: By the proof of 3.1.3.1 above, P - aPy is a point and 6 = Py e A4 because 
p ± y with p,y e Q. So we have a J_5, 0|6 because 0|P,y. This yields A,0\a,b 
with ^ == a6 so that A = Ohy Axiom 3. ■ 

3.1.3.3. If A e X and 5 e V, then A ^ 5; that is, a point does not equal a plane. 
Proof: Let A = a^ with a e ^, P e A^, and a ± p. Suppose that ^ = aP = 5 e P. If 
b & M then we have a = P6 e ^ with P|5 and p,6 e M, which contradicts Axiom 
5. If 5 e Q, then A = a^ & Q and this contradicts the definition of a point. ■ 

3.1.3.4. The elements ofX are involutory. 

Proof: Let A e X, so that we may write ^ = ap with alp. In particular, aP = Pa 

and AA = aPaP = aapp = 1©. ■ ^ :• . . 

. , „. .: *- "■'■'.:■' -.^h. •■ 

3.1.4. General Consequences of the Axioms '^*.-'. *' 

3.1.4.1. ///'|a,p andaL^ then P = ap andifP ^ ap thenP\(x,'^ andaL^. 
Proof: If P|a,p and aip then aP = ^ for some A & X and we have A,P\<x,^ with 
alp. Thus, by Axiom Z.A^P. 

HP = ap, then by 3.1.3.4 a I p. because P i. 7^ by 3.1.3.3 then alp. Also 
Pa = P and Pp = a imply that the products Pa and Pp are involutory so that 
P|a,p. ■ 

3. 1.4.2. IfP I a then Pa e V; that is, Pa is a plane p and P \ p. 



33 
Proof: By Axiom 1 there exists P such that P \ p and P 1 a. Thus, P | a, P with a i P 
so by 3.1.4.1 above, P = ap and P = Pa. ■ 

3.1.4.3. For each P e X and each ^ e i3, P^ e X. 

Proof: By the definition of a point we may write P = a^ with a e M , ^ e Q, and 
alp. By 3.1.2.2, a^ip^ and P^\a^,^^ so that P^ = a^p^ is a point. ■ 

3.1.4.4. //i'|a,p andjLa,^ then a = ^.( Given a point P and a plane ^ there exists 
a unique plane a such that P \ a and a _L y. ) 

Proof: This follows immediately from Axiom 1. ■ 

3.1.4.5. There do not exist three planes pairwise absolutely perpendicular. 
Proof: Suppose that a, p, and y are pairwise absolutely perpendicular. Then for 
f = aP we have P | a, P with a, P Jl y so that a = p, which contradicts a J_ p. ■ 

3.1.4.6. IfA,B,C\a then ABC = D|a. 

Proof: By Axiom 8, ABC is a point D and Da = ABCa = oABC = aD. ■ 

^.l.A.7. IfA,B,C\a,^ thenABC\a.^. 

Remark. From 3.1.4.6 and 3.1.4.7 above the product of three coplanar (and as we 

shall see, collinear) points yields a point which is coplanar (collinear) with the other 

three. ; " ' . 

3.1.4.8. //ai,a2,a3±a then a laja^ = a4 e V and a^ La. 

Proof: Let A = aai, B = aa2, and C = aas. Then by Axiom 8, ABC is a point D 

and D = aaiaa2aa3 = aaiaias. So D|a, by 3.1.4.6. By 3.1.4.2, 

aia2a3 = Da = a4 e V, D = aa4, and aia2a3 = a4±a. ■ 

3.1.5. Perpendicular Plane Theorems 

3.1.5.1. If a 1 P; Py = ^ andA\a then a 1 y. fA plane perpendicular to one of two 

absolutely perpendicular planes, and passing through their point of intersection, is 

perpendicular to both.) 

Proof: From our assumptions above it follows that ay = a^A = ^Aa = ya, so a i y 



34 
or a -L Y ( iiote that a = y implies that a i P because A - ^y and A\a). Suppose that 
a Xy so that B = ay. Then we have A,B\a,y with a ly, which impUes that ^ = 5 by 
Axiom 3. But ii A - B then Py = ay and P = a, which contradicts P -L a. Thus, ay is 
a plane and a ± y. ■ 

3.1.5.2. Suppose that a ± P; ^|a,p,y,5; yla; and 5±p. Then 5 J. y. {If two planes 
are perpendicular, their absolutely perpendicular planes at any point of their 
intersection are perpendicular.) . ,; 

Proof: By 3.1.4.1, ^ = ya = 6p and 5y = ap e P as a 1 p. Thus, 5 1 y. ■ 

3.1.5.3. IfO = aai =yyi = SS| with a,y,s e M. and a J. y J. s J. a then ay = s. 
Proof: Because a,y,s e M, then ai,yi,Si e Q. Because a J. y J. s ± a and 
aai = yyi = £8i, then ya = yiai; sa = Siaj; sy = £iyi imply that 

yi -L ai -L 8] 1 yi. Because 0|ai,yi,8i, then O = aiyi8i by 3.1.3.2. Because points 
are involutions by 3.1.3.4, loj = 00 = OaiyiSi = Oa\jiOOe\ - aye and ay = s. ■ 

3.1.6. Parallel Planes. 

We say that two planes a and P are parallel, denoted by a || p, if a = p, or 
there exists a y such that a, pj-y. 

3.1.6.1. Parallelism is an equivalence relation on the set of planes V. 
Proof: That the relation is reflexive and symmetric is clear. For transitivity suppose 
that a II P and P || y where a,p, and y are distinct. Then there exists 5,8 e P such 
that a, pis and P,yls. Let A = a5. B = P5, and C = P8. Then by Axiom 8 we have 
D = ABC = a55pp8 = as and ai.8 so that a || y. ■ 
3.1.6.2 If a,^ld and ^le then ale. 

3.1.6.3. IfaJ-j, pl5, andy \\ 6 then a || p. (Two planes absolutely perpendicular to 
two parallel planes are parallel) 

Proof: If y = 5 then we have a, pi5 and the result follows. So assume that there is 
an 8 such that y,6l8. Then by 3.1.6.2 above, ai5, ply, a,pls, and a jj p. ■ 



35 

3.1.6.4. If a II p, y || 5, and^Lb then aly. (Two planes parallel to two absolutely 
perpendicular planes are absolutely perpendicular.) 

Proof: because a || p then a, P±s for some s and because P-L5 then aX5. Similarly, 
y,6±s' for some s' and 5Xp imply that yip. Hence, a,pX5 and P_Ly yield a_Ly. ■ 

3.1.6.5. If a e M{Q) and a \\ p then p e G{M). 

Proof: Suppose that a e ^ and a || P, so a, p±y for some y. Because a e Q then 
y e tVI, which implies that p e ^. ■ 

3.1.6.6. If a II p thena^ \\ p^ for every % e 0. 

Proof: Let a,p±y for some y e P. By 3.1.2.2, a^,p^±y^ so that a^ || P^. From 
3.1.6.5 and 3.1.6.6 above and the invariance of Q and M, we have that if a || P and 
a e g{M) then p e g{M) and a^ || p^ with alp' e g{M). ■ 

3.1.6.7. If a II p then Xa n .^p = ora = p. 

Proof: Suppose that a,P±y and P\a,^. Then a = P by 3.1.4.4. ■ 

3.1.6.8. If A * B then A j( B; that is, AB ^ BA. 

Proof: By Axiom 6 there exists a e V such that A,B\a. By 3.1.4.1 and 3.1.4.2 we 

may write A = aa\ and B = aa2. Suppose that AB = BA. Then we have 

aia2 = a2ai so that aiXa2 or ai ± aj. But ai || a2 because ai,a2ia so that 

either ai = aj or .^ai (^ ^02 = 0. If ai 1 a2 then by Axiom 7, Xai^Xp ji^ so 

ai = a2 and ^ = 5. If ai J_a2 then C = aia2 e 3Cai 1^ 3602 ^^^ again we have that 

aj = aj. Hence, A ^ B. ■ 

3.1.6.9.a. IfA'^ = A, then B = A. 

h. If A ^ B, then A, B, and A^ are pairwise distinct. ■ 
3.1.6.10. IfA\a; 5|p; C|y; and a || p || y then ABC \a^y. 
Proof: Let Aa = a', 5p = p', Cy = y'. Then a',p',y'la,p,y and 
ABC = aa'pp'yy' = a'p'y' = 5'5 with a'p'y' = S'lS = apy. ■ 
Conclusion: A\a; 5 1 P; a || p, imply that 5' | p. 



36 

3.1.6.11. IfAA' = BB'; A,A'\a; B\^; a \\ p then B' \^. 

Proof: Because^' - BB' then B' = BAA'; p || a || a. From 3.1.6.10 above, 
B' = BAA' \^aa = p. ■ 

3.1.6.12. For each P and each P there Is a unique a such that P\a and a \\ p. 
Proof: By Axiom 1 there exists a unique y such that yip. because P|y then Pj = a, 
aiy and a || p. Now ii P\8 and 6 || p then PI 6, a, so that a = 6 by 3.1.6.7. ■ 
3.1.6.12. Leta,^ e V and M e X. Then aip <^ a^'ip and thus, a || a^. 

Proof: Suppose that aXp. By Axiom 1 there exists unique y,5|M such that aXy and 
5lp. By 3.1.6.2 we have p,yia and pl5 which implies that yi5. By 3.1.4.1, 
M = y6. thus a-^' = a^'^ = a^lp^ - p. Therefore, a'^ip and a^ || a. 

Conversely, suppose that a-^^lp. As above, there exists unique y,6|jV/such 
that a^iy and 5ip. It follows that pi5, M = y6, and a^ = a^^ = a'^ip and 
a = (aS)SipS = p. ■ 

li A-^^ = B, then we say that M is the midpoint of A and B. Clearly, M is also 
the midpoint of B and A. 

3.1.6.14. IfP'* = P'^ then A ^ B. ( Uniqueness of midpoints.) 

Proof: From P^ = P'^ we have PAB = ABP and also PAB = PBA because ABP is a 
point and hence, an involution. Thus, AB = BA which implies that ^ = 5 by 3.1.6.8. 
■ 

3.1.6.15. IfA,B\a,^ andA^ = B then M\a,^. 

Proof: By Axiom 6 there exists y,5 such that A,M\y,5- because B = A'^ = MAM then 
by 3.1.4.7 we have 5|y,5. Thus, ^,5|a,y,5 ; M|y,6; and ^,.e|P,y,5 so by Axiom2, 
A/|p,a. ■ 

3.1.6.16. .IfA,B,C\S; a,p,yl5; A\a; 5|p; C|y; a || p || y thenABC\a^y and 
apyl6. 

Proof: From 3.1.6.10 we know that D = ABC\a^y = s. Because 5|a,p,y then e|5 so 
that 8 1 6 or eld. If £ = 5s then we have D,E\5,£ with Sis which implies that 



37 
D = E hy Axiom 3. Let A = aa', B = pp', and C = yy'. Because a || P || y then 
a' II p' II y'. Thus, s5 = Z) = ABC = aPya'p'y' = ea'p'y' which implies that 
6 = ot'P'y'. Because A \ 5 then we may write A = aa' = 56' and by 3.1.5.1 we have 
A\a',a,b; aXa'; and a _L 6 which impUes that a' ± 5. Hence, a'5 = a'a'p'y' = P'y' 
is a plane, so P' -L y'. By Axiom 7 and 3.1.6.7 we have P' = y' so that a' - 5. But 
this yields ai6 and a J_ 8, which is a contradiction. Thus, aPy ± 5. ■ 

3.1.6.17. ///i,B I a; /I I a': a'±a; 5|p; and^ \\ a' then ^ 1 a. 
Proof: By 3.1.6.16 above B = AAB\a'a'^ = p and p 1 a. ■ 

3.1.6.18. //£|a,£; alp, andel^ then s 1 a. 

Proof: If E\ P then the result follows from 3.1.6.7. Suppose that £ / P and let 
M = sp. Then, £ ^ £P and £P | aP = a so that E,E^ \ a. Now E^ = f'^P = £P, so that 
Mis the midpoint of £ and £P. E^^ = £P'=|a'= so that £,£^|a,a'= and by 3.1.6.15, 
M\a,a^. In particular, M|a and we have Mja, p,s; a _L P; and pis so that a J. e 
by 3.1.6.7. ■ 

3.1.7. Consequences of Axiom 11 and 3.1.6.18. 

3.1.7.1. //^|a; a -L p; and a,^ e Q then there exists ay in Q such that A\y and 
y 1 P,a. 

Proof: Let ^ 1 6 with 5lp. Then 6 6 A^ and .4 | a, 6 with alp, pl5 so that a 1 5 
by 3.1.6.18. Because 5 e M, 6 1a. and a e Q then s = 5a e Q, A\z, and s 1 a. 
We claim that sip. Indeed, Pe - P5a = 6pa = 6aP = sP with s,p e ^ so that 
e 1 p or s = p. But s 7t p because then 6a = p and a = 5p = P, as 6ip. This 
contradicts 3.1.3.3. ■ ' ; 

3.1.7.2. Every point may be written as a product of three mutually perpendicular 
planes from Q. 

Proof: Let A be any point. Then by definition A = a^ for some a e Q and some 

P e TW with aip. By Axiom 11 there exists y e ^ such that y 1 a. By 3.1.7.1 there 



38 
exists a 6 e ^ such that ^|6 and 5 ± a,y. Again by 3.1.7.1 there exists an e e ^ 
such that A\z and s ± 5,a. Hence, A \ E,a,5; s,6.a e Q, and e J. 5 1 a J_ s which 
yields by 3.1.3.2, ^ = 5aE. ■ 

We note that in this case, aP = ^ = a5£ impUes that P = 6z; that is, if ^ | P 
and P e A1 then there exists Pi,P2 e Q such that /i|Pi,P2 and P = PiP2- 

3.1.7.3. If A I a then there Js a P in Q such that A\ P and P -L a. 

Proof: This follows directly from the proof of 3.1.7.2, for if a e ^ then we can find 
5,E e ^ such that ^ |5,e and 6,e -L a. If a e M. then we can find ai,a2 e Q such 
that .4|ai,a2 and ai,a2 J- a. That is ii A\a then there exists pi,P2 e Q such that 
^|Pi,P2 and Pi Jl P2. Moreover, if a e A^ then we can find Pi,P2 such that 
a = P1P2 and if a e ^ then we can find P1.P2 e G such that aiPip2. ■ 

3.1.7.4. If A I a e A^ and P J. a then there is ay such that A\y andy J. P,a. 
Proof: Let 5|^ with 5lp. Then 6 J. a by 3.1.6.18 and A\z = a5. Because a 1 P and 
5J_P then E|p. If eXP then we would have ^|5,e with 5,sJlp so 5 = £ and a = 1©. 
Thus, £ 1 p. We note that if p e ^ then 6,£ e M and if p e A^ then £ e ^. ■ 

3.2. Lines and Planes 
3.2.1. General Theorems and Definitions. 

3.2.1.1. ForanyA,B e X and any a, b e g, ifA,B € a,b then A = B or a = b. Hence, 
by Axiom 6, for all A,B e X. there exists a unique g e Q, such that A, B e g. 
Proof: Let a = [a,p], b = [y,5], ^,5|a,p,y,5, and suppose that A jt B. Let C e a, so 
that C|a,p. By Axiom 2 it follows that C|7,5 so C € b and a <^ b. Similarly, b (^ a 
and a = b. ■ 

3.2.1.2. Every line contains at least three points. 

Proof: By definition, every line g = gAB contains at least two points A and B. Let 
gAB = [a,p] with A,B\a,^. Then by Axiom 8 and 3.1.4.7, ^*|a,p and A^ e X. 
A'^ ^A by 3.1.6.9. ■ 



39 

3.2.1.3. Suppose that ap = y. 

a. If aa i = PP i then a i -L P i and aipi=Y, 50 yj_ai, Pi. 

b. If aa\ = PPi = TYi then a ± Pi, ai J. p, yi .L ai,Pi, anda^\ = aiP = y. 

c. If A I a, P then A\y. 

d. The line g = [a, p] = [p,y] = [a,y] = [a, P,y] Is a nonisotropic line. Thus, a,^,and 
y are tJiree mutually perpendicular planes which intersect in a line. 
Proof: a. From aai = PPi we have y = pa = Pitti. . 

b. If PPi = yyi and aai = yyi then yi = yPPi = aPi and yi = yaai = Pai. 

c. Because /4 1 a, p and ap = y then Ay = Aa^ = a^A = yA and A \ y. 

d. By Axiom 7 there exist points A and B such that A,B\a,^ and by 3.2. 1. 3. c. above, 
A,B\y. Thus, A,B & [a,P],[a,y],[p,y] and [a,p] = [a,y] = [p,y] = [a,p,y] is a 
nonisotropic hne. ■ 

3.2.1.4. If A and B are collinear then A^ and B^ are collinear for any t, e 0. 
Proof: This follows from the fact that v4,5| a, p ^^^,5^ |a^,p'. ■ 

For each ^ € 0, we define a^ = {C^ : C e a}. By 3.2.1.4 above, a^ is a line 
for every ^ in © and if « = [a,p] then a^ = [a^,P^]. 

Definition of parallel lines and planes. We say the line a is parallel to the 
plane a, denoted by a \\ a, if there exists a P such that a c 3Cp and P || a; that is, 
^ = [P'Y] ^iid a, P 1 6 for some 5. 

We say that two lines a and b are parallel, a \\ b, if there exists a,P,y,5 such 
that a = [a,y], b = [p,6], where a || P and y || 5. 

3.2.1.5. IfAA' = BB' ^ 1©; A,A'\o.,o!\ a ^ a'; 5.B'|p,p'; p ^ p', then g^^' \\ ggg>. 
Proof: Let fi|p* with p* || a (3.1.6.12). Then fi'|p' by 3.1.6.11. Let B\^\, with 

Pt II ai. Then 5'ip; by 3.1.6.11 and 5,5'|p,Pi,p',p^ which implies that 

[P.Pi] = [P%P|] = gBB' by 3.2.1.1. Hence, we have g^^i = [a,ai], ggg' = [p*,Pt] with 

a II p* and ai || Pi; thus, g^^< \\ ggg>. ■ 



40 

3.2.1.6. Two lines, a - [a,ai] and b - [p,Pi] are parallel precisely when there exist 
A, A' e a and B,B' e b such that AA' = BB' * l^. 

Proof: Suppose that a \\ b. Then, a || P and ai || Pj. Let A, A' e a and B e b, so 
that B' = AA'B\aa^, and aiaipi = p,pi, by 3.1.6.10. Thus, B,B' e b with 
AA' = BB' . li AA' = BB' = 1& then A = A' . ■ 

3.2.1.7. If a II b and b \\ c then a \\ c. 

Proof: Let a = [a,a']. From the proof of 3.2.1.5 above we may write b - [P,p'] with 
a II P and a' || p' because a \\ b, and c = [7,7'], where y || p and y' || P' because 
b II c. By 3.1.6.1, a || y and a' || y' so that a || c. ■ 

3.2.1.8. For each line a and point A there is a unique line b such that A e b and 






Proof: Let B,C e ahe distinct and if A e a we can choose B,C ^ A because every 

hne contains at least three points by 3.2.1.2. Then ABC = D by Axiom 10 and 

BC = AD * ig, so gAD II a by 3.2.1.6. Now suppose that A & c and c \\ a. By 3.2.1.7 

above, c \\ gAD, so there exist W,X e c and 7,2 e gAD such that WX = YZ ^ \& by 

3.2.1.6. By 3.1.4.7, ^' = ^fFX e c and ^1 = /iFZ e ^.4Z). Thus, 

lg3 ^AA' = WX = YZ = AAx impUes that^' = Ax. Rence, A,A' e gAD,c; A ^ A' , 

because WX ^ 1© and gAD = c by 3.2.1.1. ■ 

3.2.1.9. IfA,B e g, A^ B, andC e h then g \\ h if and only if ABC eh. 
Proof: If ^5C = H e h, then 1© ^ AB = HC and g \\ h. Conversely, suppose that 
g II h and put D = ABC. Then because A i^ B, 

1© ^ AB = DC and C e gDC, with gDc \\ g- 

Because ^ || A and C e h, then by 3.2.1.8, gDc = h and D e h. M 

3.2.1.10. If a II b then a = boranh = 0. 

Proof: Suppose a \\ b and a r^ b ^ 0. Let A e a,b and C e a with C ^ A, B e b, 
with A it B. Then, a = gAc, b = gAB, and therefore, D = ACB e a,b. 



■".'T- '■■ •-,."=y'-"i " 



41 
U D = A, then ACB = A, C = B, and a = b. If D ^^ A, then it follows that 

« = gAD = gAC == ^BD = gAB = b. M 

Classification of nonisotropic lines. Following the terminology of physics we 
make the following definitions. Let o be a nonisotropic line. If there are elements 
a,p,y e M such that a = Py and a = [a, P,y], then we say that a is timelike. If there 
are elements a, P e ^ such a 9^ P and a = [a, p,a6], then we say that a is spacelike. 

Remark. Let A,B\a e Q with A ^ B. Then by Axiom 14, there is a P such 
that ^,5| P and P ± a. Thus, a = [a, P] is a spacelike line by definition, A,B e a, 
and every pair of points in a spacelike plane is joinable with a spacelike line. 



■■,.>»' 



^'■" t 



3.2.2. Isotropic Lines. f ; „ ' }*.v 

3.2.2.1. If A ^ B; A,B\a,^ with P ± a, a e M; and A and B are unjoinable with P; 
P\a,y; andjLa,^, then A'' = B. 

Proof: First, /i^|a^,P^ = a, p. U A^ = A then A\y and A is unjoinable with P. 
Similarly, 5^|a,p and B^ ^ B. Suppose that A'' and P are joinable. Then there are 5 
and s such that P,A''\8,e and 5 1s. But then P,A = /'T,(.4Y)r|6V,8T and 6^ 1 s^, 
which implies that P and A are joinable. Thus, A"^ and P are unjoinable, so by Axiom 
13, A"^ is unjoinable with A or A'^ is unjoinable with B or A"^ = A or A'' = B. Hence, 

Ay = B. m 

3.2.2.2. Suppose that A ^ B; A,B\a,^ with a L ^.^ a e M; A and B are unjoinable 
withP\a; P|y, yip. then A^ = B. 

Proof: First observe that because P\a,y; alp; and yip, then a 1 y by 3.1.6.18. 
Then A^la^, p^ = a,p and A^ is joinable with B. A^ ^ A because A and P are 
unjoinable as in 3.2.2.1 above. ^^ joinable with P implies that A = (A^y is joinable 
with py = P. Hence, by Axiom 13, Ay ^ B. M 

3.2.2.3. If A and B are unjoinable then X e g^B precisely when X = A orX^BorX 
is unjoinable with both A and B. 



42 
Proof: By Axiom 6, gAB = [a. P] for some a and p in V. Let X e g^B and suppose 
that X ^ A.B. If X is joinable with A then there exists y,6 such that 7X8 and 
A,X\^,b. By Axiom 2, 5|y,5 and A and B are joinable. Hence, Xis unjoinable with 
both A and fi. ■ 

Remark. If Xis unjoinable with J and B; A,B\a.,^; then by Axiom 10, ^|a,P 
and X e gAB- 

3.2.2.4. If gAB is isotropic; C,D e gAB', and C ^ D. then C and D are unjoinable. 
Proof: If C,D|a,p with a ± P then A,B\a,\i by Axiom 2; that is, gAB = gCD- ^ 

3.2.2.5. IfA,B,C\a are pairwise joinable and distinct, then each P\a is joinable with 
at least one ofA,B, and C. 

Proof: If a e ^ then the result follows from Axiom 12, so assume that a e A^. By 
Axiom 13 there exist D,E\a such that A,B, and C are all unjoinable with P. Then 
by Axiom 13, at least two of A,B, and C must lie on one oi gpo and gpE. By 3.2.2.4 
above, this implies that two of A,B, and C are unjoinable, which contradicts our 
assumption. ■ 

3.2.2.6. If gAB is isotropic then g\g is isotropic for all '% e <S> . 

Proof: If.4^,5^|a,p with alp then ^,5|a^"',p^"' and a^"' 1 p^"'. ■ 

3.2.2.7. IfP\a e M then there are at most two isotropic lines in Xa through P. 
Proof: If gAP,gBP,gcP (= '^"a are three distinct isotropic lines through P, then P is 
unjoinable with A,B, and C. The points A,B,C must be pairwise distinct and are 
joinable by Axiom 13, in contradiction to 3.2.2.5. ■ 

3.2.2.8. By Axiom 13 and 3. 2. 2. 7, for each P\ a, a e M, there are precisely two 
isotropic lines in Xa through P. . , . .. 

3.2.2.9. Let gAB,gAC <= -^a, a & M, be two isotropic lines. 

a. If A I p. with pla, then g^^g = gAc- 

b. IfA\y, withyla, then g^g =gAC- 



43 
Proof: Because g,4B * gAC by assumption, then B is joinable with C by Axiom 13. 
The results follow ftom 3.2.2.1 and 3.2.2.2. ■ 

3.2.2.10. IfA,B,C\a, a e M; gAB '*■ Sac are isotropic then ggc is nonisotropic and 
there is a^ with ^\A such that P 1 a and C = B^. 

Proof: Because g^B "*■ gAC^ then gBC is nonisotropic. By Axiom 14 there exists a y, 
y|5,C, such that y ± a. Let ^ | P, P 1 y (Axiom 1). Because ^ | a, P; a ± y; and 
ply, then by 3.1.6.18, a -L p. If B^=B then B\ P and g^g is nonisotropic. Similarly, 
CP ^ C and by 3.2.2.1, ^P - C. ■ 

3.2.2.11. IfA,B,C\a. a e M. are pairwise unjoinable then there exist ^,y ± a with 
A\^,y such thatC = B^y. 

Proof: First observe that by 3.2.2.3 and 3.2.2.4, 

gAB = gBC = gAC C= -^a 

is isotropic. By 3.1.7.3 and 3.1.7.4, there exist ^\A such that P -L a. Because B and 
C are unjoinable with A, then 5,C / p and B^ ^ B. By Axiom 1 and 3.1.6.18, there 
is a 6|P such that 5 1 P and 5 1 a. Thus, 5,5P|5,5P = 6,a and B and fiP are 
joinable. Suppose that ^P and C are unjoinable. By 3.2.2.3, either gg^(^ = g4c or A is 
joinable with 5P. But iiA,B^\£, s 1 a, then ^ = A^,B\e^ 1 a^ = a imply that /4 
and B are joinable. If g^p^^ = g^c = gAB then ^P is unjoinable with B. Hence, ^P and 
C are joinable and by 3.2.2.10, there is a y with y\A such that y 1 a and C = fiP^ . ■ 

3.2.2.12. LetP\a,^,y with p,y 1 a, a e M. If gAB <= -"^a is isotropic then g% c Xa 
is isotropic and gAB II g%. 

Proof: By 3.2.2.8, there are precisely two isotropic lines in Xa through P, say gpc 
and gPQ. By 3.2.2.9, g^pc = gpg and g% = gpc- Now PAB = Z) is a point by Axiom 
8, so AB = PD and D is unjoinable with P as ^ is unjoinable with B. (Suppose that 
P,D\a' with a' 1 a. Let 5|X, with X \\ a'. Then because B\a, X 1 a by 3.1.6.17. 
But then by 3.1.6.16, A = PDB\a'a'X = ?. and ^ is joinable with B.) Because 



44 

A,B,P\a then D = PAB\a. Hence, D e ^/-c or D e gpg, and g^a || gpc or 

, By II By m By m Py n h 

gAB II g/^e- thus g% II g^^ = gPQ II g^fi or g^Jj II gpc - g/>C II gAB- ■ 

3.3. A Reduction to Two Dimensions 

In this section we restrict our attention to two dimensions. In this way we are 
able to use the work of Wolff [30] to construct our field. 

Let ^ be any element in 0; then the map a^ : X >-^ X given by C7f(^) s ^£,~' 
is bijective and maps lines onto lines and planes onto planes by 3.2.1.4,5,6 and 
3.1.2.3. Hence, it is a colHneation of (0,^,X). In the following, for any element 
4 e 0, the collineation induced by it is denoted by ajc. 

Let a e P be fixed throughout this section. We wish to define a set Ca of 
maps on Xa which can be viewed as line reflections in a given plane Xa- We then 
show that each element of Ca is involutory and that Ca forms an invariant system of 
generators within the group Ga it generates. Finally we will show that {Ca,Ga.) 
satisfy Wolff's axioms [30] for a two dimensional Minkowski space; that is; the 
Lorentz plane, ior a e M.. 

Let Ca - {g <^ Xa '■ g is nonisotropic}. By Axiom 14, each g e Ca may be 
written uniquely in the form g = [a, P,y] where a = Py. Let A e Xa and 
g = [a,p,y] e £«• because A \ a and a = py, then A = ^" = A^'' and A^ - A^. Hence, 
we define the map Cg : Xa >—>■ Xa by ag(A) = A^ = A^ = A^ , for A e Xa, and ■ 
ag : £a ^ Ca by (5g{h) = h^ = {ag{A) : A e h). Note that by 3.2.1.4, if , 
h = [a,5,s], then Ggih) = [a,6P,sP] = [a,6^si']. 

3.3.1. For each g = [a,p,y] e £«, <yg '■ Xa '—>■ Xa is bijective. 

Proof: liA,B\a and ag(A) = OgiB) then A'' = B^ and A = B, so Cg is injective. If y4]a 
then ^^ ] a^ = a because a ± y and Cg(A^) = {A''y = A, so Og is onto. ■ 

3.3.2. Forg = [a,p,y] e £«, cig : Ca '->■ Ca Js bijective. 

Proof: Let h = [a,s,r|] e £«• Because a 1 y and s 1 r| <-> e^' 1 r|^ by 3.1.2.1 then 



45 
a = a^ = s^ri^. It follows that Cgih) - [a,8^,Ti^] e £«• If ^|e,ti then ^^|s^,ti^ so 
Cg(A) e C!g{h) for all A eh and Og is a well-defined collineation. 

If / = [a,A.,n] e Ca and Cg{h) - dgil) then for every A in h we have 

CTg(.4) ^Aye Ggil) = [a,X\^y]. 

This implies that A = (/i''')''' e [a,A,,[i] = /;that is, h - I. Hence, Cg : Ca '-' Ca is 
injective. Because s ± r\ <r^ z'^ 1 r\'' then it follows that 

h = [a,£,r|] e Ca <r^ Og(h) = [a,s^TiT] e £«• 



: -'•<, 



Hence, CTg : £a '— *■ Ca is surjective. ■ 

3.3.3. Each a g, for g e Ca, is involutory. 

Proof: Let g = [a,p,y]. Then for ^ | a we have OgCSg(A) = ag(A'') ^ (A^y = A. ■ 

Let g = [a,5,y], h = [a,s,r|] e Ca- We say that g is perpendicular to or 
orthogonal to h, denoted g -L h, if one of y and 5 is absolutely perpendicular to one 
of s and r| and gn h ^ 0. 

3.3.4. Forg and h as above, ifyle then 5±r|, 5 1s. andy 1 s. Moreover, 
A/= yc = 5r| e g,h. 

Proof: Let M = ye. Because y,s 1 a then Af- = (ys)" = y"s" = ys = M and M\a. So 
we have M\a,e which implies that M|aE = r| and M e h. Also, M\a,y so M\ay = 5 
and Meg. Also from M = ys it follows that Ms = y = 6a = 5ti£ so that M = 8r\ and 
5iTi. Applying 3.1.6.18 to M|y,Ti with y 1 5, Til5, and M|6,s with Sly, sly, we 
obtain y 1 r| and 5 1 e. ■ 

3.3.5. Ifa,b,g e Ca then a 1 b <^ a^ 1 b^. 
Proof: By 3.1.2.1 and 3.1.2.2 we have 

a c Xy- yl5; b ^ Xs <r^ a" cz X^; y^15^; b'^ c xl, for all ^ e (5. 
The result follows. ■ 



46 

Remark. Because A\s <r^ A^\z^ for all ^ e 0, A e X, and s e 'P, it follows 
that for each g e Ca. CTg maps: 
(i) The set Xa onto itself. 
(ii) The set £a onto itself. 

(iii) Collinear points in Xa onto collinear points in .^a- 
(iv) orthogonal lines in Xa onto orthogonal lines in Xa- 
That is, Gg is an orthogonal collineation of Xa- 

Let Ca = {cTg : ^ e £«} and *Pa = {a/> : P e Xa}- 

3.3.6. The sets Ca and Xa are nonempty. Hence, Ca ^ and'^a "*■ 0. 
Proof: If a e Al then we may write a = a'5 where a', 5 e Q and a' ± 6 by 
definition of M.. By Axiom 7, there exist A,B such that A ^ B and A,B\a',b. Thus, 
A,B\a'6 = a and Xa ^ 0- Moreover, g^B - [a, a', 5] e Ca and Ca "^ 0- li ct. e Q 
then by 3.1.3.1, there exist A\a and by 3.1.7.3, there is a p in ^ such that A\P and 
P ± a. Thus, A e Xa and g = [a,p,aP] e Ca- ■ 

Note that from 3.3.3, Ca consists of involutory elements and by 3.1.3.4, ^a 
consists of involutory elements. 

3.3.7. IfP e g,h e Ca and g 1 h then Op = OgOh = ^h^g- 

Proof: Let g = [a, 7,5] and h = [a,8,ri], and without loss of generality, assume that 
yls, so that 5ir|. By 3.3.4, P = ye = dr\ e g,h and by 3.2.1.1, {P} = gnh. Then 
for A e Xa, 

ogOhiA) = Gg(A^) = A^y = Gp(A) = Ay^ = GhiAy) = uhGgiA). m 

3.3.8. For every A In Xa and for every h e Ca, there is a unique g e Ca such that 
A e g and g 1 h. 

Proof: Let h = [a,s,Ti] e £« and A e Xa- By Axiom 1, there exist ay\A such that 
yls. Thus, A\a,y with a 1 s and yls so a 1 y by 3.1.6.18. because a 1 y then 
ay = 8 and A\a,y implies ^ 1 5 so that A e g = [a,y,5] e Ca- because g c Xy, yls. 



47 
A e g = [a, 7,5] e Cn- Because g c Xy, yJ-S, and h c Xe, then g -L h. By 3.3.4 
above, tiX5 and {B} = {ys} = {t|5} = gnh. 

Now suppose that I a Ca such that A e I and I 1. h. Now we may uniquely 
write / = [a,a',P] in jEq where a = a'p. because I ± h then without loss of 
generality, we may assume that a'_L£ and P-Lr|. because y-Le and 81.r\ it follows that 
a' II y and P jj 5. By the definition of parallel lines in Section 3.2 we have / || g. 
because /i e /ngthen by 3.2.1.10, I = g. ■ 

3.3.9. (i) The fixed points of a g e Ca are the points Jn Xa incident with g. 
(ii) The fixed lines ofog e Ca consist ofg and all lines in Ca which are 

orthogonal to g. 

Proof: Let g = [a,y,6]. li A - C!g{A) - A^ for A \ a, then A \ y. because A'' = A^, ^A \ a 

then A^ =AandA\ 5. Thus, A e g and 3.3.9. (i) follows. 

Let h - [a,s,ri] € £« and suppose that Ogih) = h; g ^ h. Then there is an 
A e h such that A € g. By 3.3.8, there is a k - [a,co,0] e £« such that A e k and 
k ± g. Because A e h, OgiA) = A'' e h and because A e k, k 1 g then 
^^ I (0^,0^ = (0,9 and ^^ e k. Because A ^ g then A )( y (for if y4|y then A\a implies 
that ^ jay = 5 which implies that A e g) and A * A^. Thus we have A,A'^ e h,k, so 
that h = khy 3.2.LL Hence, h 1 g. M 

3.3.10. (i) The only fixed point of a point reHection, Gp, is the point P. 

(ii) The fixed lines ofop are the lines incident with P. 
Proof: If P' = P, then this implies that APA = A, or AP = l<s, or A = P. Thus, 
3.3. 10. (i) holds. Let x = [p,u] be any line (not necessarily in Xa or nonisotropic) and 
A e X. Then P-^ e x implies that P-^ \ p,u, so that Fj p'^,u-< = p,u, and P e x. M 

Let Ga be the group acting on Xa generated by Ca- By 3.3.5, every element of 
Ga is an orthogonal collineation of .^a- Let ct e ^a- By a transformation with a, we 
mean the adjoint action of or on ^a : on e Ga ^ of = ctctiCT"' e ^a.Note that every 
such transformation is an inner automorphism of the group. 



48 

3.3.11. Let h = [a,E,ri], g = [a, 7,6] e £«, then Gg* = (s^^(g). 
Proof: Let A \ a and put B = OgiA) = A'^. Then 

Thus, (Tg*(a/j(^)) = ffCT/,(g)(cf;i(^)) for all A e .^a- Because a>, is injective, then 
CTg'' = (To^{g) for every A in ^a- Similarly, C!g''(a) = ao^,(g)(o), for every a in Ca- 
Hence, dg'' - Oai,(g)- Analogously, we can obtain ct^'' = a^^^^p), for every P in Xa- B 
Because Ca generates Ga and every element of Qa is an orthogonal coUineation 
of {Xa,JO.a) then we obtain the following. 

3.3.12. For every (J e Qa and for every Og e Ca, we have ct| e Ca', that is, Ca /5 a/2 
invariant system of Q a- B 

Consider the two mappings: g e Ca ^-^ ^g ^ Ca and P e .^a *— >• <7/' e ^a- 
The first one is from the set of nonisotropic lines in £« onto the set of line 
reflections, and the second is from the set of points in .^a onto the set of point 
reflections in Xa- These mappings are injective, because reflections in two distinct 
points have distinct sets of fixed points by 3.3.10, and similarly for lines and line 
reflections. Thus, it follows from 3.3.11, 3.3.12, and the preceding remark that the 
next result obtains. , - - 

3.3.13. //a e Ga and P,Q e Xa then aiP) = Q <^ af = cjq. 
Proof: Because 

the result follows. ■ 

3.3.14. Iffy e Qa and g,h e £« then G(g) = h <r^ a^ = o/j. ■ 

3.3.15. IfP e Xa andg e £« then P e g <^ apCg is involutory. 
Proof: Suppose that P & g= [a,y,6] e £„■ Then for .4 | a, 

cpdgiA) = (yp(A'') = Af^ = A'^ = OgGpiA). Thus, {cpOgy = l.i-„. Now assume that 



49 
apug is involutory. Then P = a gO pg gcy p(P) = P^^^ = P^'^. Because yPy ^ P^ e Xa 
by 3.1.4.3, then by 3.1.6.9, P = P^. Because P'^ = P^ then P\y,d, and hence, Peg. 

m 

3.3.16. Ifg,h e Ca, then g 1 h <r^ CgOh is involutory. 

Proof: Nowg ± /? if and only if ag(h) = h. For g ^ h, by 3.3.9, Cgih) = h li and only 
if a^^ = a/,. For Gg -^ u/, by 3.3. 14, a'^/ = a/, if and only if (agG/,)^ = Ixa and 
GgGh ^ l.^a- ■ 

3.3.17. TJie point reflections ^a are Z^ize involutory products of two line reflections 
from Ca- 

Proof: Let ap e ^a where P e Xa- Then we may write P = ySs where a = y5 and 
y,6,s e ^ by 3.1.7.3. Let g = [a,y,5]. Because a = y5, then g e Ca- By 3.3.8, there 
is an / in £« such that Pel and I -i- g. By 3.3.7, Gp = CT/CTg = <7gCJ/. ■ 

We now show that if a e A1, then the pair (Ca,Ga), acting as maps on 
(j£a,'Ca,a), satisfies Wolff's axiom system [30] for his construction of 
two-dimensional Minkowski space. We give Wolffs axiom system below. 
Basic assumption: A given group &, and its generating set Q of involution elements, 
form invariant system iG,<3) 

The elements of Q will be denoted by lowercase Latin letters. Those involutory 
elements of (3 that can be represented as the product of two elements in Q, ab with 
a I b, will be denoted by uppercase Latin letters. 
Axiom 1. For each P and for each g, there is an I with P,g\l. 
Axiom 2. IfP,Q\g,l then P = Q or g = I. 
Axiom 3. IfP\a,b,c then abc e G. 
Axiom 4. Ifg\a,b,c then abc e Q. 

Axiom 5. There exist Q,g,h such that g\h but Q / g,h,gh. 

Axiom 6. There exist A, B; A ^ B, such that A, B ^ g for any g e Q. (There exist 
unjoinable points.) 



50 
Axiom 7. For each P and A,B,C such that A,B,C\g, there Is a v e Q such that P,A \ v 
orP.B\v orP,C\v. 

Geometric meaning of the axioms. The elements denoted by small Latin 
letters (elements of Q) are called lines and those elements denoted by large Latin 
letters (thus the element ab with a\b), points. We say the point A and the line b are 
incident li A\b; the lines a and b are perpendicular {orthogonal ), \{ a\b. Further we 
say two points A and B are joinable when there is a line g such that A,B\g. 

Replacing £« with Ca and Xa with ^a, the pair {Ca,Qa) satisfies the basic 
assumption by 3.3.3, 3.3.6, and 3.3.12. It follows from 3.3.15, 3.3.16, and 3.3.17, that 
our definition of points, our incidence relation, and our definition of orthogonality 
agree with those of Wolff. Hence, we may identify £« with Ca, Ca with Q, and Qa 
with 0. 

Verification of Wolff's axioms. Axiom 1 follows from 3.3.8 and Axiom 2 fi:om 
3.2.1.1. For Axiom 3, let a = [a,a',s], b = [a,p,p'], c = [a,Y,y'] € Q. Then 
a',p,y 1 a, and by our Axiom 9, a'Py = 6 1 a and d = [a,5,a5] e £„• For yi|a, 

(SaObCciA) = ^tP«' = ^«'Pt =A^ = Gd(A). 

Hence, CaObOc = cj^ e Ca. If we then identify "|" with "e", we get Wolffs Axiom 3. 

Axiom 4. Ifa,b,c\g, then abc e Q. , ,. -i* „ . . '\''' 

Proof: Let g = [a,XXl a = [a,a',5], b = [a,p,p'], and c = [a, 7,7'], with 
a',p,7 1 X. Then by 3.1.4.8, a'py = z L X. Let A e a, B & b, and C e c, then 
^|a',a; 5|a,p; and C|a,7. By 3.1.6.10 /15C = D|a'p7 = e and .45C = D|a by 
3.1.4.6. Thus, Z)|E,a; z LX; a L X, and s 1 a. So, d = [a,s,as] e £«• And if X|a, 
then UaOhCciX) = ^P«' = J^ = a^(A) and UaObGc = a^ e Ca M 

Axiom 5: There exist Q,g,h such that g\h but Q / g,h,gh. 



51 
Proof: Because a e M, we may write a - Py, where P,y e Q and p J- y. By 3.2.1.2 
every line contains at least three distinct points. Let g - [a,p,y] e Ca and let 
fi e g. By 3.3.8, there is an h in Ca such that B in h and h ± g. By 3.2.1.2, there is 
an A in h such that A ^ B. By 3.3.8, there is an / e Ca such that A e I and I L h. 
By 3.2.1.2, there exists Q e I such that Q ^ A. 

Now a Q e h, then /i,(^ e l,h implies that Q = A or I = h, so Q H h. Suppose 
that Q e g. Let Q e m. m J- I, so that cq = diGm- Because ag = GgO/,, 04 - CT/,ct/, 
and BAQ - E e Xa, then 

<JE = '^B<^A<^Q - ^g^h^h^l^t^m = ^g^m and g -L W. 

Because Q e g,m and g 1 w then cr^ = <jgCSm = O/Gm and Og = Cj ov g = /. This 

implies A,B e Um. Thus A = B or I = m, a contradiction. ■ 

Axiom 6: There existA,B;A ^ B, such that A, B / g for any g e Q. 

Proof: This follows from our Axiom 13. ■ 

Axiom 7: For each P and each A,B,C such thatA,B,C\g, there is av inQ such that 

P,A\v. orP,B\v, orP,C\v. 

Proof: By our Axiom 13, there exist two isotropic lines in .^a through P, say gpg 

and gPK. If no such v in £« satisfies the above, then two of the three points must lie 

on one of the isotropic lines by our Axiom 13. But this implies that gpg = g or 

gPR = g', that is, g is isotropic; a contradiction. ■ 

3.4. Consequences of Section 3.3 
3.4.1. For every A and B, there exist a e M such thatA,B\a. {Every line lies in 
some Minkowskian plane.) 

Proof: By Axiom 6, there exist a & V such that A,B\a. II a e M, the result 
follows. So suppose that a e ^. By Axiom 12, there exist p e ^ such that A,B\^ 
and p ± a. Then A,B\a^ = y, and y e >( by the definition of M. ■ 



52 

3.4.2. For every A and B, there exists M such that A^ = B; that Js, every two points 
has a midpoint and by 3.1.6.14, the midpoint is unique. 

Proof: By 3.4.1 above, there exists a g A^ such that A,B\a. From Section 3.3, there 
exists M\a such that A^ = B. ■ 

3.4.3. IfAA' = BB' then a and A' are joinable precisely when B and B' are. 
Proof: Suppose that A and A' are joinable and let A, A' |a,a' with a -L a'. By 3.4.2, 
there exists an M such that A''^ = B. Then B = A^\a^,a"^ and from 3.1.2.1 and 
3.1.6.13 it follows that a^ 1 a'^, a || a-^', and a' || a'^. By 3.1.6.11, 

AA' = BB'; A,A'\a,a'; B\a^,a'^; a \\ a^'; and a' || a'^. 

Thus, B' I a^,a'^ and, B and B' are joinable. ■ 

3.4.4. If a. 1 a'; a || P; a' || p'; and [a,a'] \\ [p,p'], tlien p 1 p'. 

Proof: Because [a,a'] |i [P,p'], then there exist A,A'; A,A' \a,a' and there exist 
B,B'; 5,5' I pp', such that AA' = BB' . Then for A^ = B, as in the proof of 3.4.3, 
[P'P'] =^BB' - [a^^a'^] II [a,a']. That is, 5|a'^^a'^p,p'; with p,a^ || a, and 
p',a'^ II a'.Thus, p = a^ and P' = a'^' by 3.1.6.12. Hence, p ± p' because 

3.5. Construction of the Field 
The basic construction. For the construction of the field we will follow the 
path of Lingenberg |20|. Throughout this section let a e M and (9|a be fixed. 
Define the sets: 

Oa = {g e Ca : O e g) and Pa(0) = {(Tga/, : g,h e Oa}. 

Proposition 3.5.1 The set T>aiO), acting on the points ofXa, is an abelian group. 
Proof: By 3.1.7.2 we may write O = ap = yrip with a = yx\; y,Ti,p e g-, and y,y\, and 
P mutually orthogonal. Thus, g = [a,y,Ti] e £„: O e g, and Oa ^ 0. Because each 



53 
Gg e Ca is involutory then Ixa ^ T^aiQ). Now let Oa^'^b^^c^d ^ T^a{0) where 
a = [a,a',aa'], b = [a,p',ap'], c = [a,y',ay'], and d - [a,5,a5] are nonisotropic. 
Now 5y'p' = E ± a with 0|s by Axiom 9 and/= [a, 8, as] e Oa- Thus, for A\a, 

Hence, GaOhOcOd = GoGj e Va(0). Because each a/ is involutory for / e Ca, 

CT^' = cj/ and {GaGhY^ = GhGa e Va{0). From Axiom 9 and the calculation above, 

the product of any three of 8,7', P', and a' is an involution. Thus, 

^8y'p'a' ^ ^P'y'Sa' ^ ^P'a'Sy' ^^^ GaGbGcGd = GcG^GaGi,. That is, VaiO) is abelian. 
Clearly, VaiO) is associative so that T>a{0) is indeed an abelian group. ■ 
Lemma 3.5.2. Let g be an isotropic line in Xa with O e g. Then for every 

GiGi, e VaiO), G/Ghig) = g. 

Proof: Let / = [a,p,aP] and h = [a,y,ay] be Hues in £« with O e l,h. By 3.2.3.12, if 

0| P,y with p,y 1 a then gPv || g. But OP^ = O so that CT/,a/(0) = O^^ - O. Thus, 

gPY =ghy 3.2. L 10. Hence, ci;,cj/((9) = g, for all (j/,a/ e VaiO). ■ 

Lemma 3.5.3. Z/e^g be an isotropic line in Xa through O. Then for every A, E e g, 

E ^ O, A * O, there is a unique g/Gi, e VaiO) such that g/G/XE) = A. 

Proof: By 3.2.3.11, there exist y,5 1 a with 0|y,5 such that £T^ = A. Take 

/ = [a,y,ay] and h = [a,6,a5]. Note that li E = A then E^^ = E implies that F' = E^ 

or E' = E^, which implies that / = h [30]. To show uniqueness, suppose that 

GaGh,GkGi e VaiO), where / = [a,5,a6], k = [a,s,a8], g = [a,y,ay], and 

h = [a,p,ap] are lines in XaiO) and E ^ E''" = £*. Then £''«*' = E and ^P^'^S = E. 

By Axiom 9, pys = X with 0\X, XI a. Thus, m = [a,X,a'k] e OalGkGaGh = Gm, and 

E = E"''. Let £| a',5' with a' Lx, 5'i6, and put M = a'X and N - 5'5. Now 

£|a',a,5' with a IX, a 1 ?t, 6' 1 5, and a ± 5, so that a', 5' 1 a by 3.1.6.18. It 

follows that M^ = a'«;t« = a'X = Mand ^« = 5'«6« = tV . So that M\a,X ■ N\a,b; 

and m - goA/ and / = goN. Then because E'"' = E, we have E^^ = E; E^ = E^ and 



54 
£A/ = ^a'x ^ £8'8 ^ £A' and M - ^ by 3.1.6.14. So w = ^o.w = ^o.v = / and a/ = a,„. 
Therefore Oa^h = ^A<^/- ^ 

Let us denote this unique map by 6^ . So for all A e go£, /I ^ O, 
(i) 6.,(0)-0 
(ii) 5^(£) = /J 
(iii) 5.4 e Pa(0). 

Translations. For every pair J, 5 of distinct points we can define a translation 
Tab '■ ^ '— *■ -^ given by T^siA) = AAB = 5. We now restrict our attention to a set of 
translations defined on .^a and we note that [iA,B,C\a, then D = ABC\a by 3.1.4.6. 
Thus, Tab '• X.a '-^ Xa is a well-defined map for all A,B\a. Let /C = goE be an 
isotropic line in Xa and define Ta = {Tqa '■ A e K,}. 
Theorem 3.5.4. The set Ta is an abelian group. 

Proof: Let Tqa^Tob e Ta. Because C = BOA = AOB e IC, for Xla, we calculate 
(ToA ° 7-05)^ = ToAiXOB) = XOBOA = XOC = Toc{X). Thus, Tqc = Tqa ° Tqb e Ta. 
Also, Too{X) = XOO = X. Hence, Too e Ta is the identity l^a on Xa- To find r5|^, 
we compute 

(ToA ° 7^0.4'^ )W = ^O^^O^ = XOOAOOA = X = XO^O^^ = {Tq^o o ToA)iX). 

Hence, 7^^ = T^ao, A^ = OAO e IC, and 7^^ e Ta. Clearly, 
ToA ° (Tob ° Tqc) = {Tqa ° Tqb) ° Toe for A, B,C e /C. Therefore the action of Ta is 
associative and Tx is a group. Because ABC is a point and hence, an involution for 
all points A,B,C, then for A,B e IC and X\a, 

iToA ° Tob)(X) - XOBOA = XOAOB = {Tqb ° ToA)iX). 

Therefore, Ta is abelian. ■ 

Lemma 3.5.5. For all Tqa e Tx, ro.4(/C) = K.. 

Proof: Because Ov45 e IC for A,B e IC then ro.4(X;) c X; for all Toa e Ta. Now let 



55 
C e IC and A e IC. Then OCA = D e )C so that C = ODA and ToAiD) = C. Hence, 
each ToA ^ %. maps ^ onto IC. M 

Lemma 3.5.6. Let Tqa e Ta and g c Xa- Then ToAig) = g if and only if g is 
parallel to /C. 

Proof: Let gnr == g be a line in Xa such that ToAigHp) = gHF- Then 
ToAiH) = HOA e gHF and g///r || K by 3.2. L9. Conversely, suppose that gHF II ^• 
Then again by 3.2. L9 it follows that for B e gHF, 

Toa(B) = BOA e gHF and ToA(g) = ^- ■ 

Lemma 3.5.7. IfA,B e h, h a Xa^ and h \\ /C, tJien there exists Tqc ^ %i such that 
Toc(A) = B. 

Proof: By 3.2.L9 we have C = OAB e /C and it follows that B = AOC = TociA). ■ 
Lemma 3.5.8. For each A e IC, there is a unique Tqa e %l such that Toa(0) = A. 
Proof: Clearly, ToAiO) = OOA = A. So suppose that TociO) = A. Then 

ooc = c = A. m 

We denote this unique translation mapping of O into A hy Ta- 
Lemma 3.5.9. If a e VaiO) and Ta e Ta, then oTao-^ = T^^y 
Proof: Let ct = OgCy, where g = [a,P,aP], h == [ay, ay] e Oa- Then for any Xja, 

By 3.2.3.8 there exist precisely two isotropic lines in Xa through O : K, = goE and 
^' = gOF- We define multiplication and addition on the points of IC so that the 
points of IC form a field. For A,B e IC, define: 
A + B ^ (Tb o TaXO) 

A • B = {dB° 6 .()(£), where A,B * O and £ e /C is the multiplicative identity. 

A'0=0'A = 0. 

Theorem 3.5.10. For every A, B e IC, Ta+b = Ta ° Tb. 



56 
Proof: For X\a, Ta+bW = Taob(X) = XOAOB = XOBOA = {Ta o TbKX). ■ 
Theorem 3.5.11. For all A,B ^ O in /C, 6^.5 = 8,4 o 6^. 
Proof: Let 5^ = CTad^' and 6a = cr^a^'. Then A-B = 658^ (£) = £«'«*'*. Put 
Gc = a^'(7„CT^', then, ct^cTc e Va{0) and a/,cjc(£) = £^* = £"'^*'* = A - B. Hence, by 

3.5.1.3, dA-B - ^b^c = ^b^b'^a^a " ^a^^h'^b^a' = f^a^o'<^6^6' = ^A^B- ■ 

Hence, (/C,+) is a group isomorphic to Ta and (/C\{0},-) is a group isomorphic 
to VaiO). It remains to show that the distributive laws hold. 
Theorem 3.5.12. Let A,B, C e IC, then (A + B) - C = A - C + B • C. 
Proof: If C = O, then 

(A+B)'C = 0=0-0+0'O^A'C + B'C. 
li C 7^ O, then we compute 
(A+B)-C = 5ciA + B) = 5cTa+b(0) = 5cTa+b^cH0) - ScTaTb^cHO) = dcTA^c^dcTBd 

= nc{A)nc(B)iO) = TA.cTB.ciO) = TA.c+B.ciO) = A'C + B-C. 
Because multiplication is commutative, 

C'iA+B) = iA+B)-C^A'C + B'C = C-A + C-B. Hence, (/C,+,.) is a field. ■ 

Ordering the field and obtaining R. To order the field K. we make use of the 
following [21]. Let F be a field and ^,,... ,/!„ e F. If /f,,... ,^„ ^ implies that 
X^=i Aj ^ 0, then F is called formally real 

f 

Theorem 3.5. 13. (Artin-Schreier) Every formally real field can be ordered. ■ 

To make the field /C formally real the following axiom is posited. 
Axiom F. (formally real axiom). LetO,E\a, a e M with O and E unjoinable. Let 
X,5,r| e V. IfO\X,b,r\ andX,b,r[ 1 a then there is ay e V such that 

0\y, y 1 a, and E'^^^^OE^'^'^'^ = £^t^y. 

Theorem 3.5.14. // Axiom F holds on K, then /Cis formally real. 

Proof: Let / = [a,?.,a?.] e Oa and let dgcs,, e Va{0). Then from Proposition 3.5.1, 



57 
CgGijG, = a I for some I e Oa and CTga/, = ct/ct,. Clearly, if /' e Oa with a^'G/ = ct/ct; 
then we have a/- = a/ and /' = /. Hence, for all agCJ/, e VaiP), there is a unique 
/ e Oa, such that cjgCT/, = CT/CT,. So if O ^ ^4 e /C, then we may uniquely write b^ in 
the form GaCT,; that is, for every O ^ A e /C, there is a unique a, a = [a,a',aa'] e Oa 
such that 

A = aaO,(E) = E'" = £^«'. 

Suppose that O^ A = £^"'. Then ^2 = £>.a'Xa' ^ ^ implies that 
£ - QaXaX ^ Q^ a contradiction. Hence, if O ?t ^i e /C then A~ ^ O. Now suppose 
that A,B e }C and ^,5 ?t O. Let o = [a,a',aa'], b = [a,p,aP] e Oa such that 
A = £^" and B = £^P. Then by Axiom F there exists a y g P such that 0|y 1 a 
and 

A^+B^ = £^a'^a'o£^P^P = E^^^y. (3.5.1) 

Since 0|y 1 a then c = [a,y,ay] e Oa,Oca, e Pa(0), and 

C = GcGtiE) = E" = E^y e /C. 

So equation (3.5. 1) reads A- + B^ = C^. If C^ = f^v^^Y = O, then E = O^'^^^ = O. 
Since E ^ O, it follows that if ^i,...,/l„ e K, are all nonzero, then Y.l=\Al * O and 
/C is formally real. ■ 

To finally obtain a field isomorphic to the real numbers, M, we add the least 
upper bound property to our axiom system. '■ 

Axiom L. 7/0 ^ AciK. and A is bounded above, then there exists an A e }C such 
that A>X, for all X e A, and ifB e KL with B > X for all X e A then A < B. 

The only ordered field up to isomorphism with the least upper bound property 
is the real number field, R. 

Theorem 3.5.15. The field IC constructed above, along with Axiom F and Axiom L, 
is isomorphic to the real number field, R. 



58 
In the next section an affine vector space is constructed from products of pairs 
of points. The scalar multiplication is obtained by adapting and extending the 
definition of multiplication of elements of K. 

3.6. Dilations and the Construction of {X.V.K) 
The additive group V and dilations. First we construct a vector space V over 
the field /C. Let V = {OX : X e X}. First note that the product, AB, of any two 
points A,B ^ X, is in V because AB = 0(OAB) = OOAB. We view the elements 
OX € V as directed line segments with initial point O and terminal point X on the 
line gox- We define an addition on V by setting OX+OY= OXOY. The product of 
three points is a point, so XOY = Z e X, OXOY = OZ e V. 
Theorem 3.6.1. (V,+) is an abelian group. 

Proof: Let X, Y,Z e X be distinct. Then, 0X+ OY = OXOY = OYOX = 0Y+ OX, and 
addition is abelian. The zero vector. is 1© since 1& - OO e V and 
1© + OX = leOX = OX = OX+ 1©. To complete the proof we calculate 

0X+ 0X0 ^ OXOXO = OXOOXO = OXXO ^00= 1©, so -OX^ oxo. 
(OX+ OY) + OZ= {OXOY)OZ = OX{OYOZ) = 0X+ {OY + OZ). 

Hence, (V ,+) is an abelian group. ■ 

We still need to define a scalar multiplication of K on V. To do this note that 
in an affine space the group of dilations with fixed point C is isomorphic to the 
multiplicative group of the field. Noting this, we geometrically construct such a 
group of mappings and use these mappings to define our scalar multiplication. 

A dilation of X is a mapping b : X ^ X which is bijective and which maps 
every line of X onto a parallel line. [23, p.37] 

Theorem 3.6.2. [23, p.42| A dilation 5 is completely determined by the images of 
two points. 



59 
Proof: Let 5 : J£ i-» JC be a dilation and assume that 5(A) = ^ and 5(7) = T' of two 
points X,Y e X are known. We must show that the image of any Z e X is known. 
Suppose that Z g gxr- Then clearly Z ^ X and Z ^ Y and we consider the two lines 
g^ and gYz- Observe that Z e gxz ^ grz- If these lines had a point in common 
besides Z, they would be equal and then gxz = grz = gXY- But this is impossible 
because Z g gxr- Therefore, {Z} = gxz f^ grz- Since 5 is bijective, from set-theoretic 
reasons alone, d(gxz ^ gvz) = ^igxz) ^ ^(grz); or equivalently, 
{5(Z)} = d(gxz) f^ ^(gvz)- In other words, the lines 5(gxz) and digyz) have precisely 
one point in common, namely, the point 5(Z) for which we are looking. The line 
8(gxz) is completely known because it is the unique line which passes through X' and 
is parallel to gxz- Similarly, the line 5(gYz) is the unique line which passes through 
the point Y' and is parallel to gYz- because 5(Z) is the unique point of intersection of 
^(gxz) and 8(gyz) (3.2.1.1), the point 6(Z) is completely determined by X* and 7*. 

Conversely, assume that Z g gxy- If Z is Jf or Y, we are given 5(Z), so assume 
that Z 1^ X and Z t^ Y. By 3.4.1, there is an a e A1 such that X, Y\ a and there exists 
a P I a such that P ^ gxy. Then Z € gxp and from the previous paragraph, d(P) is 
known. Hence, using the line gxp instead of the line gxy, we conclude from the 
earlier proof that 5(Z) is known. ■ 

To define a scalar multiplication, we fix a timelike line / and use it to 
geometrically define dilations. To aid in the construction, the following facts are 
used to add the appropriate axioms. 

In a three-dimensional or four-dimensional Minkowski space, if / is any 
timelike line through a point O and g is any other line through O, then there is a 
unique Lorentz plane containing the two lines. Two distinct isotropic lines 
intersecting in a point in Minkowski space determine a unique Lorentz plane. 
Desargue's axiom, D, holds in any affine space of dimension d > 3. 



60 
Axiom T. IfO e t,g where t is timelike or t and g are both isotropic then is a unique 
(X e. M. such that g,t a Xa- 

Axiom D. Let g\, gj, and gj, be any three distinct lines, not necessarily coplanar, 
which intersect in a point O. Let P\,Q\ e g\; P2,Q2 e g2; and Pt„Qj, e ^3. // 
SP\Pi II gQxQi ^ndgp^p^ II gQ^Q^ thengp^p^ \\ gg^g^. 

Axiom R. LetO e gugi; P\,Q\ e gi; and Pj.Qi e g2. Ifgp^p, \\ gg^g^ then 
gO,P\OP2 = go.Q^OQi- 

Axiom T refers to the first statements. Axiom T is used to put an isomorphic 
copy of the field on every isotropic hue through O. A scalar multiplication is then 
defined in a manner similar to the definition of multiplication for the field elements. 
Axiom D is Desargue's axiom, the "dilation" axiom. Axiom D ensures well-defined 
dilations with the standard properies of such maps. Axiom R is used to distribute a 
scalar over the sum of two vectors. The dilations are constructed next. 

Let IC c Xa, a & M.By 3.1.7.2, there exist ai,a2 e Q such that 0|ai,a2 
and a = aia2. Because Oja, then Oa = p e ^ and O = a^ - aia2P with o.\,a2 X p 
by 3.1.6.18. Let y = aiP e M. Then ya = Paiaia2 = Pa2 e M, so that 
t = [a,y,ay] c Xa is timelike and O e / as 0|a,ai,a2,P implies that 0|a,y,ay. So 
let t = [a,y,6] e Oa be any timelike fine through O and )C' c Xa the other isotropic 
line through O. Then for each At e t, there is a unique A e K, and there is a unique 
B e K.' such that At = AOB. Because Ai e t, then B ^ A'. (From this point on 
denote a line reflection GgiX) by X^. Thus, X^'"' means arCf,Og(X) = agCi,c!r(X).). 

For each A, e /, there is a unique A e K. such that At = AOA'. Similarly, for 
each A in IC, there is a unique At in / such that At = AOA'. Thus, there is a 
one-to-one correspondence between the points of IC, the field, and the points of /. 

Fix Et - EOE' as the unit point on /. For each O ^ A e IC, we use t to 
construct a dilation 6"^ of X in the following way. Let X e X. 



61 
If X /, then by Axiom T, there is a unique r\ e M. such that goxJ <^ -^ti- 
From Sections 3.5 and 3.6, j£ti is an affme plane and our definition of parallel lines in 
^n is equivalent to the affine definition. So, there is a unique line h c: Xri such that 
A, e h and h || gxE,- Because 

goxnt = {O} ^0, hnt = {A,} * 0, and h \\ gE,x, 

then hngox = {B} * 0. In this case, set E^iX) = B. 

li X & t and X ^ O, then because t,K, a Xa with a e M, there exists a 

unique g (z Xa such that A & g and g \\ gxE- Because gxE r\t = {E] t- then 

i 

^n/= {fi} ^ 0. Set 6^(A) = fi. ^, .SV 

If X = O, set hA{X) = O. Note that 5^(A) e ^oj^ by construction. It is clear 
that 5^ : ^ H^. X is one-to-one. We find it useful to make some observations. 

Let O ^ ^ e /C and recall that £ e /C is the multiplicative unit point; 
E ' A = A, io^ A ^ K. VvX At = AOAK 

IjQmm&^.Q.Z. For the map b A defined above, the following are true: 

1 ?>a{E)=A. 

2 ?>A{E,) = At. 

3 ^A is onto. 

Proof: I. E, = EOE' and A, = AOA' imply that EEi = OE' and AAt = OA'. If 
EE, = OE' = le then O = E' and O = O' = E. Similarly, because A ^ O then 
AA, = OA' ^ U. By 3.2.2.9, A' = Ay e K,' as 0,A e /C, /C is isotropic, and 0|y 1 a. 
Similarly, E' e K,' . Thus by 3.2.1.6 we have gEE, II goE' = IC' || g.,,,, and gEE, II g^^, 
by 3.2.1.7. By 3.2.1.8, h = gAA, and h n goE = gAA, r^ JC = {A}. Hence 5^(£) = A. 

2 5^(£,) = ^,. Again, g8^(^E)A, = gAA, II gEE, and g^^, n / = {A,}. 

3 Let P e X and P ^ O. U P ^ t then by 3.2.1.8, there is a unique fine g such that 
Et e g, g II ^4,p, and gngop = {Q}, for some Q e X. Then it follows that 

5^(0 = P.UPet then 5a(Q) = P, where {^} = ^n /, £ e g, and g \\ gAP. ■ 



62 
Lemma 3.6.4. IfC ¥■ D, then gco II ^6^(C)8^(Z3)' hence, for all O ^ A e K, 6^ is a 
dilation ofX. 

Proof: Let g\ = t, g2 = goc- gi = gOD, P\ = Et, Q\ = A2 & g\. Pi = C, Q2 = 5^(0, 
Qi G ^2, ^3 = A and ^3 = 5..)(£>)- Then ^c£, II .?^,8^(o and gz)£, || gA,h^(Dy Thus, by 
Axiom D, gcD II gs^(08^{Z))- ■ 

Consider now the plane Xa- Recall that by 3.5. L3, iov O ^ A e IC, there is a 
unique 64 e VaiO) such that 5^(£) - A, where 6^ is of the form GgC^f, for g,h e Oa- 
From Axiom 9 and the proof of 3.5. LI, for any r e Oa, agCT/,CTr = Gw for a unique 
H- 6 Oa. That is, for every fixed r e Oa, every O ^ A e IC can be uniquely writen in 
the form 5^ = 0^^^ where Gw(E'') = A. (The uniqueness follows from the fact that 
(Ay^ = iA')''^ implies w^ ^ wj ii A' ^^ A [30].) 

Therefore, for every O ^ A e IC, there is a unique a e Oa, such that E'" = A. 
That is, there is a unique a e Oa, such that a^CT, = 5.,). Also, every P e Xa, P\oi, 
can be uniquely writen as P = P1OT2, where P\ e IC and P2 e IC'. 

Let P2 = P2' ^ fC. Then we may uniquely write P = P\0P2, where P\,P2 e /C. 
Define the map 6^ : .Xa ^ -^a, by 5^(P) = P'CO(P'f)', hr O ^ A e IC, P\a. 
Theorem 3.6.5. The map 8a : Xa ^ Xa is a dilation on Xa with fixed point O and 
dilation factor A, for all O ^ A e /C. 

Proof: If P'^OPf = QfOQ'f, then by unicity, Pf = Qf. So Pi = Q\, P'f = Q'f, 
and P2 = Qi- Hence, P ^ PiOP'2 = QiOQ'j = Q, and 5^ is injective. Let g|a with 
Q = Q\OQ'2, QuQ2 e K. Let P, = (?f e /C and P2 = 0' e IC. Then P = FiOP^Ia 
and 6^(P) = PfOPf = iQfy^OiQfY"' = Q,OQ\ - ^. Therefore, 6^ is onto. 

We claim that li P ^ O, then gpg || g%^(p)%^(Qy First we show that 
^AiP) e go/'. If P 6 /C, then P = /'OO' and S^C?) = P'^OO"" - P'«00 = /"^ e AC. If 
P e IC', then putting R ^ P' e IC, we have P = 007?' and dA{P) = O'^OR"" = 
= OOR"" = /?'"' = P"' e IC'. 



63 
So suppose that gop - g is nonisotropic and write P = PiOPj, where P\ = P'f- 
UO e g, then XOY e g<^X= YS, where X e JC and Y e K.' . Thus, 5^(P) = PfOP'f 
and P'f = Pf = Pf^, which impUes that 5a(P) e g. 

Claim. dA(POQ) = 5^(^)05^(0, for ah P,Q e Xa- Let P = P^OP'2 and Q = Q\OQ\. 
Then POQ = P\OP\OQxOQ^2 = {.P\OQ\)0{P20Q2y, with P\OQx e /C and 
{P2OQ2)' e fC'. Therefore, 

^a(POQ) = (PxOQxY^OiPiOQj)''' = PfOQfOP'fOQ'f = (PfOP'f')O(QfOQf) 
= dA(P)05A(Q). 

The claim follows. 

Proof of (iii): Assume that P,0, and Q are collinear. Then there is a line g, such that 

P,0,Q e g. Thus, 6^(P),6^(0 g g and gpQ = g \\ g = g^^^p)i,^^Qy 

Conversely, suppose that P, Q, and O are noncollinear. Let g\ = gop, gi = gOQ, 
and g2, = go.POQ- Then g\,g2, and ^3 are distinct lines through O. Indeed, if ^3 = gi, 
say, then POQ e gop and we would have OP(POQ) = Q e gop, which contradicts 
our assumption of noncoUinearity. Now, 

PiPOQ) = OQ and 6^(P)(5,4(/'O0) = 6^(6^(/')O6,,(0) = 06^(0 

gPJ>OQ II gOQ = go6^(Q) II gh^iP)8^iPOQy ^nd gp^poQ || g8^(/>)5^(/.O0- 

Similarly, ^(^'00 = ^(eOP) = OP; and 6^(0(6^ (/'O0) = 6^ (6,4 (006^ (P)) = 05^(P). 

This implies that gg^pog \\ gop and gop = goh^p) II ^5^(05^(PO0- Hence, 
gQ,POQ II g5^(06^(/>o0- By Axiom D, it follows that gpg \\ g^^^p)^^(^Qy Therefore, 
5a '■ Xa >—^ Xa is a dilation on Xa- B 

Now we show that 5^ = 6^ on Xa- Because 6^ and "6^ are dilations on aCa, a 
dilation is uniquely determined by the images of two points,and 6^(0) = O = 5^(0), 
then it suffices to show that 6,4 (£) = 6^(£). By definition of 6^, 5a(E) = A. By 
Lemma 3.6.3, 6^(£) = A. 



64 
To extend the above idea to any r| e A^ such that t c Xr\, let a 9^ r) e M 
such that / c Xr\. Let IC\ and AC2 be the isotropic Hnes in Xr] through O. Because / is 
nonisotropic, then by Axiom 14 there exist y,6 such that r) = y5 and / = [r|,y,5]. Thus 
(Tr : X,^ i-^- Xri is well-defined. Every B e t a Xr\ may be uniquely written as 
B = B\0B2, where 5, e /C,. Because O e t and t is nonisotropic, then B2 = B\. Thus, 
B = B]OB2 where B] & IC\. So, in particular, there are unique E\,A\ e IC\ such that 
Et = E\OE\ and At = A\OA\, for At e /.As before, there is a unique a\ e Ot) such 
that E"^ = A\. Hence, every X e X^ can be uniquely written as X - X1OX2, where 
X\,X2 e )C\ d Xr]. This defines a map 6^,^ : X^ 1— >• X^, given by 

dAr^iX) = X'C'OX'p' , for X e Xr^. 

Proposition 3.6.6. Let ^A^ : Xr[ ^ X^ be the map defined above. Then 

1. 6^,1 is a dilation on Xr\. 

2. 6^,1 = 5^ on X,y 

3. ifP,0, and Q are collinear points in Xr\, not necessarily distinct, then 

bAy,{POQ) = 8Ar,(P)Od4r,(Q). 

4. Moreover, Ia{POQ) = h(P)05AiQ), for every P,Q\y\ such that 0,P, and Q are 

collinear. U 

To obtain a scalar multiplication on V, for all O =^ A e IC and all OX e V, 
define 



A-OX= 6.4 (0)5,4 (A) = OdA(X) and 6> • OX = 1 



». 



We now verify the vector space properties. 

Lemma 3.6.7. If A, A' e /C and OP e V, then (A + A') - OP = A - OP + A' - OP. 

Proof: Suppose P e / and P ^ O, and recall /,/C c Xa. Let P,E\a,^; A\y with y || p; 

A'\y\ y' II p; and 0|5 with 6 || p. Then y || y' || 6; g, = [a,y] || gEP with g, c= Xa, 

82 = [a,y'] II gEP with ^2 e .^a; and ^3 = [a, 6] || gEp with ^3 c .^a- Thus, 

bA{P) e gint and d^iP) e g2 n t. Thus, dA(P)\a,y and 5^iP)\a,y'. It follows that 



65 
dAiP)Od^iP)\a,y8y'; 5^(P)05^-(/') e t- y6y' = c || p; and ^O^' la^ySy' = s- Therefore, 
AOA' e [a,s] || ggp and 5^(P)06^'(P) e m [a,8]. Hence, B^oA'i^ = h(P)05^>iP). 

Suppose P ^ t. Because gopJ <^ ^ri, then replacing E with Ei, A with At, A' 
with /4„ and a with t] in the first part of the proof and the result follows. ■ 
Lemma 3.6.8. If A e IC and OP, OQ e V then A - (OP + OQ) = A • OP + A - OQ. 
Proof: We need to show that dA(POQ) = 5^(P)O6.4(0. Suppose P,0, and Q are 
coUinear. Let g = gop = goQ = goj'OQ and r\ & M such that g,t c ?i^. Then the 
result follows from Proposition 3.6.6(iv). 

Conversely, suppose that P, O, and Q are not collinear. Because P, O, and Q are 
not collinear then P ^ Q, and from Lemma 3.6.4 it follows that gpg \\ gg /mg ,Qy 
Applying Axiom R we obtain go,POQ = go5A{P)h(Qy '^^at is, 6^(/')O5^(0 e go,poQ 
by construction. Again by Lemma 3.6.4, 

8h(P)UPOQ) II 80J>0Q and a.4(P)6.4(P)O5.,(0 = 05^(0. 

This implies that g5^(/>),5,(/>)o6,(0 II ^08,(0 = gOQ II gp,POQ, as P(POQ) = OQ. Thus, 

SdA(P),h(P)05A{Q) ^ .?5^(P)8^(/'O0 because both lines contain 5,j(P) and are parallel to 

gP.POQ- Since 

S5AP).h(P)08AQ)^SP.POQ = {5..,(/')C>5^(0} and g5A(P)h{POQ) ^ go.POQ = {5^(/'O0}, 

then 5^(P)O5.,(0 = 5^(PO0. ■ 

Lemma 3.6.9. IfA,A' e K and OP e V then A • (/!' • OP) = {A -A') • OP. 

Proof: We need to show that 5^(5^'(P)) = 5^.^'(P); that is, 5.^ o 5^- = 5^.^-. Now 

5^ o 5^' and 6^.^/ are dilations on X and "5^.^'(0 = O by construction, so it suffices 

to show that (6,4 o 5.,0(£) = ^A.A'iE) and (5.^ o 5^,){0) = 5^.^'(0). Now £ e /C c JCa, 

so on .^a, 5.4 ° 5.,' = 5.4 o 5^, and 5^,y = S^.^-. But on /C, by Lemma 3.5.11, 

5.4 ° 5^' = 5^.^'. Therefore, (5.4 o 3^>)(E) = A ■ (A' - E) = (A - A') • E = d^.^^^. ■ 

Lemma 3.6.10. For E e /C. the multiplicative unit, and OP e V. E - OP == OP. 



66 
Proof: We need to show that dg = l.x- Now Ix is clearly a dilation on X and 
1 ^(O) = o = 6£(0). Thus, lx{E) = E = E'E = dsiE) = 5e(E). ■ 
Theorem 3.6.11. The space (V,)Q constructed above Is a vector space. ■ 

The triple (X,V,/C), is an affine space. [23, p. 6] A set X along with a vector 
space V over a field /C is an afBne space if for every v e V and for every X e 3£, 
there is defined a point vX e X such that the following conditions hold. 

1. If v,!^ e V and X e X, then (v + w)X = v(wX). 

2. If denotes the zero vector, OX = X for all X e X. 

3. For every ordered pair (X, Y) of points of X, there is one and only one vector v e V 

such that vX = Y. 

The dimension n of the vector space V is also called the dimension of the affine space 

X. 

Theorem 3.6.12. (X,V,IC) is an affine space. 

Proof: If OV,OW e V and X e X, we have 

(i) {pV)X=OVX e X. 

(ii) (PV + OW)X = {OVOW)X = OV(OWX). 

(iii) 00X= 1<&X = X. 

(iv) for Y e X, OYX ^Z e X and iOZ)X = Y. 

Now if OPX = Y, then OPX = OZX, P = Z, and OP = OZ. ■ 

3.7 Subspaces and Dimensions 
In this section we show that our lines and planes have the proper dimensions. 
We are then able to conclude that (V,/C) and (ie,V,/C) are four-dimensional spaces. 
Proposition 3.7.1 Let g be any line through O and put giO) = {OA : A e g}. Then 
giO) Js a one dimensional subspace ofV. 

Proof: First note that !& = OO e g{0), so the zero vector is in g(0). Let A,B e g. 
Then C = AOB e g hy 3.1.4.7 and OA + OB = OAOB ^ OC e giO). From Section 3.6, 



67 
5^(5) e g, for all A in K. and all B in g. So A • OB e g(0) for all ^ e /C and 
OB e g{0). Hence, g(0) is a subspace of V. 

It must now be shown that the dimension of g{0) is one. li g = t, then 
giO) = (OEt) because for every At e t, Ai - 6^(£;) by Lemma 3.6.3. So suppose that 
g =^ t and fix B ^ g. Let h - gBE,- Then for slX O ^ D & g, there ia a unique rfsuch 
that D e d, d \\ h, and drM ^ 0. Put {Ft] = dnt. Then for F, = FOF' with F e IC 
it follows that 

6f-(5) = D and OD = F - OB. 

Hence, ^O) = <05). ■ 

Corollary 3.7.2. Following the terminology of Snapper and Troyer (23, p.U], 

g = S{0,g{0)) ^{A = 0{0A) : OA e g(0)} 

is an afSne subspace of dimension one. 

Proposition 3.7.3. Let a e V with 0|a and put tai.0) = {OA : A\a}. Then taiO) is 

a two dimensional subspace ofV. 

Proof: Clearly, 1© = OO e taiO), so the zero vector is in ta(0). Let C,D|a and 

A.B e IC. Then by 3.L4.7 we have COD = F|a and by Lemma 3.6.3, 

5/1 (O e goc cz Xa and 6b(D) e goo c= Xa 

so that dA(QOdBiD)\a. It follows that OC + OD = OCOD = OF e ta(0), and 

A-OC + B'OD = O'dAiQ + OdBiD) = 0(6^(0055(D) e .ta(0). 

Hence, ^aiO) is a subspace of V. 

Thus, it remains to show that ta(0) is two dimensional. There are two cases: 
a e >1 and a e g. Suppose first that a e A^. We construct a basis for ta(0) using 
isotropic lines. To this end, let /Ci and /C2 be the isotropic lines in .^a through O. For 
any Pla we may uniquely write in Xa, P = PxOPj, with Pi e /Ci and P2 e Ki- 



68 
From Proposition 3.7.1 above, ^i(O) = {OB) for any O ^ 5 e /Ci and K.2(,0) = (OQ 
for any O t^ C e ICj. From this it follows that P] = 6^(5) and Pj = §^'(0 for some 
A, A' e /C. Thus, 

OP = A -08 + A' 'OC 

and {OC,OB} span ta{0). Now \i A - OB + A' - OC = 1©, then C>5^(5)06^'(Q = I©. 
This implies that 6^(fl)05^'(C) = O and Ol^^{C) = 5^(5)0. Because 0,5^'(C) e /C2 
and 0,5^(5) e /C,, then either /Ci || Kj or (95^'(Q = 1© = 6^(fi)0. But /Ci ]f /C2, so 
5^'(0 = 5.4(5) = O. Because a e A1, then from 3.3 and 3.6 there exists t' ,a e Oa 
such that 5^'(0 = C'"' = O. This implies that C = O"''' = O ov A' = O. By 
assumption C * O and D ^ O, thus it must be the case that A = A' = O. Hence, 
{OC,OB} is a linearly independent set in %a(,0) and -ta(0) is two dimensional. 

Suppose now that a e Q. We construct an "orthogonal" basis. By 3.1.7.2, there 
exist p,Y e ^ such that O = apy and a 1 P 1 y 1 a. Let jc = [a,p,aP] and 
y = [a,y,ay]. Note that if ;c = _v then by Axiom 14, aP = y and O = aPy = 1®, so 
x^y. Let P| a and let p',y'|/'with p' 1 p and y' 1 y. Put P, = pp' and P2 = yy'. 
NowP|a,p'; a 1 p; p' 1 p, so that a 1 p' by 3.1.6.18. Because P| a, y'; a 1 y; 
y' 1 y, then a 1 y'. Thus, 

P? = (PP')« = P«P'« = pp' = P, and Pf = (yy')« = yy' = Pj, 

so P\,P2\a.. This implies that P\ |a,P and P2|cx,y so P] ex and P2 e y. If 
Q = PxOP2 then Q ^ P\OP2 = P'ppayyy' = p'ay'. Because p' 1 a, let § = ap' e V. 
Then Q = 5y' and 5 1 y'. Thus P| p',a implies that P| p'a = 5. Thus we have 
P,Q\d,y' with 6 1 y', and P = ^ by Axiom 3. That is, P = P1OP2 with P] e x and 
P2 ey. 

Let O ^ X e X and O :^ Y e y. Then from Proposition 3.7.1 above we have 
X = (OX) and y = (OY) and there exist A,B e IC such that OP = A • 0X+ B • 07, where 



69 
Pi = 6a(X) and P2 = 8b{Y). Hence, {OX,OY} spans .ta(0). li A • OX+ B - OY ^ 1©, 
then we obtain O5.4(J0 = ^b(Y)0. Because x J[ >», it follows that ^^{X) = O = 85(7)- 
Let ^ti be the unique plane containing t and ;c and K,\ and K.2 the isotropic lines in 
Xri through O . Then we may write §4(^0 = X'^OX'^', where X^ e /Ci, and X^ e IC2. 
As above it follows that A = O and similarly, B = O. Therefore, ^aiP) is two 
dimensional. ■ 

Corollary 3.7.4. Xa = S(0,!^a(0)) Is an affme subspace of dimension two for all 
a e P, 0|a. 

Theorem 3.7.5. (V,/C) is a four-dimensional vector space and lience, (X,V,}Q is a four- 
dimensional affine space. 

Proof: Let OP e V and let O = ap with a e X and p e ^. Let P|a',p' with a' 1 a 
and p' 1 p. Put P' = a'a and P" = p'p. Then Q = P'OP" = a'aappp' = a'p'. Thus, 
P,e|a',p' with a' 1 p'. Therefore P = Q = P'OP" with P'\aand P" | p. Since taiO) 
and .tp(0) are two-dimensional then there exist bases {OZ,OT} c .ta(0) and 
{OX,OY} c= tp(0) such that 

OP' =A'OZ + A' • OT and OP" = B • 0^-+ B' • 6>y 

for some ^,^',5,5' e IC. Thus, P = ^ -02 + y4' -OT+B-OX+B' • 07 and 

{OX,OY,OZ,OT} span V. If ^ • 0Z + ^' • 07+5 • (9X+ 5' • 07 = I©, then in 

particular, OP = OP' + OP" = 1©. This implies that OP" = P'O. So either 

Sop" II ^/"o or P" = P' = O. But ^^p" if gop' and therefore, 

A '0Z + A' -07= OP' = 1& = OP" = B'OX+B' 'OY. As was shown in Proposition 

3.7.2, we obtain ^ = ^' = 5 = 5' = O and the result follows. ■ 

3.8 Orthogonality 

In this section we extend the definition of orthogonality to include lines and 
then use this definition to define orthogonal vectors. 



70 
Definition 3.8.1. Let g and h be two lines. We say that g is perpendicular to or 
orthogonal to h, denoted by g J. /?, if there exist a,p e "P such that g c Xa, h c Xp, 
alp, and P = a^ ^ g,h. In this case, P is the point of intersection of g and h. 
Lemma 3.8.2, Ifg and h are isotropic then g is not orthogonal to h. 
Proof: This follows directly from our definition above and from our definition of 
alp. For if a 1 P then one of a and p must be in Q and by Axiom 12, all lines in a 
plane J£p for P e Q are nonisotropic. ■ 

In Minkowski space, if g and h are two isotropic lines then g L h ■(r^ g \\ h. 
Thus we extend the above definition in the following way. 
Definition 3.8.3. If g and h are isotropic lines, then g L h ■(r^ g \\ h . 
Definition 3.8.4. If g is a line and a e P then we say that g is orthogonal to or 
perpendicular to Xa, g 1 -^a, if there exists a P e T' such that g (= JEp, P 1 a, and 
P = ap e g. In this case, P = ap is the point of intersection of g and Xa- 
Definition 3.8.5. For every 1® ^ OAMB e V, we say that OA is orthogonal to OB, 
OA 1 OB, if and only if go^ 1 gos; that is, there exist a,p e P such that goA c Xa, 
gOB c Xp, and O = ap. For the zero vector 1(5= OO, we define !<& L OA, for all 
OA e V. 

Lemma 3.8.6. Fronj 3.1.2.1 and 3.1.2.2 it follows that for?, e 6: 
{ij g 1 /? <^ g^ 1 h^. 
(ii) gl Xa "^g^ 1 X^t. 
(Hi) OA lOB^ (OA)^ 1 (OB)^. ■ 

Lemma 3.8.7. Ifg is a nonisotropic line then g is not orthogonal to g. 
Proof: If g 1 g then there exist a,\^ e V such that g c Xa,Xp; alp, and 
P = ap e g. But for every point Q e g, Q\a,^ with alp which implies that P = Q 
by Axiom 3; that is, g is a line which contains only one point, which contradicts the 
definition of a line. ■ 



71 
Additional axioms and their immediate consequences. To complete our 
preparations for defining our polarity and thus obtaining the Minkowski metric, we 
recall our final three axioms. 

Axiom U. (U^ subspace axiom) Let 0,A,B, and C be any four, not necessarily 
distinct, points witli A,0\a; 0,5|P; 0,C|y,6 and a 1 y and ^ 1 5. Then there exists 
?.,s e V sucli thatXl e; OAOB\'k\ and 0,C\z. 

Axiom SI. Ifg cz Xa, a e ^, /z c .tp, (3 e Q, and there exists y,5 e V such that 
y 1 5; yd e gnh; g cz Xy-, and h c .^5 then there exists e e g such that g,h c Xg. (If 
g and h are two orthogonal spacelilie lines then there is a spacelike plane containing 
them.) 

Axiom S2. Let g and h be two distinct lines such that P e gnh but there does not 
exist ^ e V such that g,h c .^p. Then either there exists a,y 6 7^ such that a 1 y, 
g (= Xa, andh c Xy or for all A e g, there exists B e h such that P and APB are 
unjoinable. 

Lemma 3.8.8. Ifg,h cz Xa are nonisotropic for a e V, then g 1 h in the sense of 
Section 3.3 if and only ifg 1 h in the sense of Definition 3.8.1. 

Proof: Let g = [a,p,6], h = [a,y,?L] c Xa with a = (35 = yX. Recall that g Ihin the 
sense of 3.3 if , without loss of generality, P 1 y and A ^ ^y e g r\ h. So, in 
particular, A = ^y with A e g n h and g cz Xp and h c Xy. So g 1 A by 3.8.1. 

Now suppose that there exist r|,s e 7^ such that B = r\e e gnh and g (^ X 
and h c Xe, but it is not the case that ply, nor that p 1 A, nor that 5 1 y, nor 
that 6 1 A. Then by 3.3.8 there exists a unique / e £„ such that B e I and 
I = [a,u,^] for some u,^ e P with a = u|i, u 1 p, and |a 1 5. If / 9^ /z then /a(5) and 
ha(B) span .ta(5). Thus ii C e g there exists L e I and H e h such that BLBH = BC. 
By Axiom U, BL 1 BC and BH 1 BC imply that BC = BLBH 1 BC. That is, gig, 
which cannot happen for nonisotropic g. Hence, I ^ h and the result follows. ■ 



72 
Lemma 3.8.9. If a e V then for each P \ a and for each nonisotropic line g c Xa 
there is a unique nonisotropic line h a Xa such that P e h and h L g. 
Proof: This follows directly from 3.3.8 and Lemma 3.8.8. ■ 

Lemma 3.8.10. Ifg,h c Xa for a e V and g,h are nonisotropic, then g -L h if and 
only if (J gO I, = CT/,ag ^ l.i-„. 

Proof: This follows from 3.3.16 and Lemma 3.8.8. ■ 

Lemma 3.8.11. Suppose that O e c,d; c,d (^ Xa, ex e V with c and d nonisotropic 
and c -L d. Let C e c and D e d, then go,DOC is not orthogonal to c ifC,D ^ O. 
Proof: First note that go.DOC "^ c ov d because then we would have DOC = D\ e d, 
say, so that C = ODD\ e d and O e d implies that C = O or c = d. Now if 
go.DOC -L c, then in Xa we have Gq = cJcG^ = (yg(,j-,^x.(Jc, which implies that 
d = gO,DOC- ■ 
Lemma 3.8.12. Let g,x c Xa with g isotropic, a e V, and g n x = {O}. Then g is 

not orthogonal to x ifg ^ x. 

Proof: If X is isotropic and x 1 g then because x ^ g, there exists y,d e V such that 
g a Xy, X c Xs, and y J_ 6. But then one of y and 5 must lie in Q. But no element in 
Q can contain an isotropic line so x is not orthogonal to g. 

Suppose that x is nonisotropic and let O e g D x. Because g is isotropic then 

a e M and by 3.3.8, there is a unique nonisotropic line h (^ Xa such that O e h and 
h Ix. Suppose that g ±x and let O ^ K e g, O ^ H e h, and O * X e x. Then OH 
1 OX and OKI OX so that by Axiom U, 0{HOK) = OHOK 1 OX. li go.HOK is 
nonisotropic then go.HOK = h. And because go,HOK c= Xa, then HOK = //, eh. Then 
we have K = OHH\ e h and g = h, a contradiction. 

Suppose that goMOK is isotropic. If goMOK = g then we may write HOK = Kx 
for some Kx e g. It follows that H = KiKO e g and g = h. li go.HOK * g then goMOK 
is the other isotropic line through O in .^a- Because g and go,HOK span Xa and 



73 
g,go.HOK -L X then by Axiom U, x is orthogonal to every Hne in Xa through O. So in 
particular, x -L h^ which implies that x = h. ■ 

Corollary 3.8.13. Ifg Js isotropic and x L g tlien either x - g or x is nonisotropic and 
X and g are noncoplanar; that is, there does not exist 8 e V such that x,g c X5. 
Lemma 3.8.14. Ifg is isotropic, x is nonisotropic, x L g with x ^ g, and {P} = x H g, 

then for all P * A eg and for all P ^ B e x, g and gAPB are not coplanar. 

Proof: From Lemma 3.8.12, x and g are noncoplanar. Suppose that A,0,AOB |y for 

some y e V. Then B = OAAOB\y, which implies that x = goB <= Xy and 

g = goA <= Xy, a contradiction to Lemma 3.8.11. ■ 

Lemma 3.8.15. IfOC e V is isotropic and OB e V is nonisotropic with OC 1 OB then 

OCOB 1 OC. 

Proof: By Lemma 3.8.13, goc and goB are not coplanar and by Lemma 3.8.14, goc 

and go,coB are not coplanar.By Axiom S2, either goc -L gocoB or there exists 

D e gocoB such that O and COD are unjoinable. So if goc is not orthogonal to gocoB 

then go.coD is isotropic and by Axiom T, there exists a unique d e V such that 

gocgo.coD c= -^6- This implies that 0,C,COD\d, D = OCCOD\8, 

gOD = go.coD c= -^8, and therefore, goc <= -Xg, a contradiction. ■ 

Lemma 3.8.16. The zero vector, 1© = 00, is the only vector orthogonal to every 

vector in V. 

Proof: This follows immediately from Lemma 3.8.11 above. ■ 

Theorem 3.8.17. IfU is a subspace ofV, then U^ = {OA e V : OA 1 OB, VOB e U} 

is a subspace ofV. 

Proof: because the zero vector 1© is orthogonal to every vector in V by definition, 

then 1© &U^. Let OA e U^ and /? e /C. because R»OA e< OA >, the subspace 

generated by OA, and goA 1 gOB for every OB e U by the definitions oW^ and 

orthogonal vectors, then R»OA e U^. 



74 
Let OA,OB e U^ and OC e U. If OC is nonisotropic then OC ^ OA,OB and 
there exist a,P,Y,6 e V such that 

A,0\a; 0,5|p; 0,C|y,6; a 1 y; and p 1 5. 

By Axiom U, there exists 'k,e e V such that O = A,s, AOB\X, and 0,C |e. Thus, 

OA + OB ^ 0{AOB) 1 OC. 

Now suppose that OC is isotropic. The following possibilities exist. 

(i) If OC * OA,OB, then as in (a) above, (OA + OB) 1 OC. 

(ii) If OC = OA = OB, then because OC is isotropic, AOB = COC e goc and 
{OA + OB) 1 OC. 

(iii) If OC = OA It OB, then OB _L OC implies that OB is nonisotropic and 
(OC+OB) = OCOB 1 OC by Lemma 3.8.14. 

Hence, if OA,OB e U^ then OA + OB e U^ and U^ is a subspace of V. ■ 

Theorem 3.8.18. IfO = ap then ta{0)^ = .tp(0) and t^{0)^ = taiO). 

Proof: From the definition of orthogonal vectors we clearly have ta(0) a .tp(O)-^ 

and tp(6>)^ c ta(0). Now suppose that OA 1 .ta(0); that is, goA 1 a. Then, there 

exists y e V such that OA e ty(0) and y 1 a. But then we obtain (9 = aP = ay so 

p = y. Hence, taiO)^ = -tp(0) and ta(0) = -tp(0)^. ■ 

An immediate consequence of Theorem 3.8.18 is the following. For each a e V 

with 0\a, there exists a unique ^ e V such thatO|p, taiO)^ = ^fi{0), and 

Theorem 3.8.19. Ifd{0) js nonisotropic then there is a unique hyperplane AiO) such 
that d{0)^ = A{0) and (fl(0)^)^ = d(0). 

Proof: Let a e V such that a a Xa, where a is the nonisotropic line associated with 
d{P) = {OA : A & a]. Then for O = ap we have a ± p and o 1 g for every g c Xp 
with O e g. Thus, .tp(0) c ^(O)-^. Now a is nonisotropic, so by 3.3.8, there exists a 
unique h c .^a such that O e /? and a 1 /z. By Axiom U, </?(0),tp(0)> c d{0)^. 



75 
Suppose l(s ^ OB e d(0) n (h(0),tpiO)).Then B e a a Xa and there exists 
OC e h(0) and there .exists OD e .tp(0) such that OB = OCOD; that is, there exists 
C e h and D \ p such that B = COD. because h c Xa, Qa and C = aa|. Because 
Z)|p, D = pp, for some p, e 7^. Now a 1 p and p 1 p,, so a || Pi. But B = COD 
and COD = ajaaPpPi = aipi so that 5|pi,a with a || pi. This imphes that a = Pi 
so D = O, B = C, and a ^ h, a contradiction. Hence, d(0) n (h{0),tpiO)) = {!©} 
and V = d(0) © </?(0),.tp(0)>. U O e d and d 1 a then diO) c </7"(0),.tp(0)>. For 
otherwise we would have 

V = d(0) ® d(0) @ </7"(0),tp(0)>, 

which is not possible. Hence, d{0)^ - </z(0),.tp(0)>. 

On the other hand, from .tp(0)^ = -ta(0), a 1 h. and a ± p, then, 
d(0) c (h(0),ta(.0))^. If O e ^ 1 p, then by definition g ^ Xa- H O e g 1 h, then 
g = a. Hence, d(0) = {h(0),XpiO)}^. 

Claim. d{0)^ is independent of the plane containing a. Suppose a cz Xy ^ Xa- 
Then for O = y5, we have y ^e a, 5 ^ p, and a 1 X?,. Again there exists a unique 
/ c Jy such that O e / and / 1 a, so that d(0)^ = </((9), -t5(0)> as above. Now any 
point P e X may be written as P = PxOPi with P] |a and Pj \ p. Since h ^ a, then we 
may write Px = //O^ with H e h and ^ e o. Then 

<MO),tp(0)> ®d{0) = V= </(0),.ty(0)> ©«(0), 

and it follows that </j"(0),.tp(0)) = </(C>),.ty(0)> = A{0). ■ 

Theorem 3.8.20. Ifd{0) Is isotropic tlien there is an unique hyperplane A{0) such 

thatdiP)^ = A{0),d{0) d A(0), andidiO)^)^ = d(0). 

Proof: Let d(0) c taiO) and a the isotropic line in Xa corresponding to d(0). 

because a is isotropic then a e A1. Let O = ap, p e ^, and put 

A(0) = (a(0),.tp(0)>. Since a c Xa, alp, and O = a^ e a, then a 1 Xp. Because 



76 
a is isotropic then a 1 a. By Axiom U and Lemma 3.8.14 it follows that 
<d(0),tp(0)> = A{0) c d{0)^. lig{0) c <d(0),tp(0)) is isotropic and g ^ a, then 
there exists a unique y & M such that g,a a Xy. Because g(0) c A{0) a d(0)^, 
then g ± o, which contradicts Lemma 3.8. U. Thus, AiO) contains no other isotropic 
line. 

Let h 1 a with O e h. For each H e h we may write H = H' OB, with //'|a and 
5|p. If O ^ ^ e o, it follows that OB 1 O^; O// 1 OA, so that 05^ 1 OA and 
O//' = 0//C>5^ 1 OA. This implies that gg^^ = a so H' e a and /2(0) e <d(0), j&p(0)>. 
Thus, fl(0)^ = <a(0),-tp(0)). 

If //'(O) c <o(0),tp(0)>^, then 

hiO) c .tp(0)^ = .ta(0) and h(0) 1 d(0). 

Thus, h(0) = diO). Hence, <d(0),.tp(0)>^ = d(0) and (d(0)^)^ = d(0). 

The subspace ^(O)-^ is independent of the plane containing a. Suppose a cz Xy, 
y e M.y^a.FutO = y6. Let h(0) c {d(0),t8(0)). Then /? 1 a and 
^((9) c (d{0),tp(0)). If 

^(O) c <d(0),.tp(0)> 
then A: 1 a and k(0) c <fl(0), ^5(0)). Therefore, 

<d(0),.t8(0)> = <d((9),tp(0)>. ■ 

Remark. If g is a line then g is either nonisotropic or isotropic. From Theorem 
3.8.19 and Theorem 3.8.20, if W < V is a one-dimensional subspace then U^ is a 
uniquely determined hyperplane. 

In an affme space a plane is uniquely determined by two distinct intersecting 
lines. Let g and h be two distinct intersecting lines and let {g,h) denote the unique 
plane determined by g and h. 



77 
Definition 3.8.21. U a e V such that g,h a Xa then we say that (g,h) = Xa is a 
noDsingular plane. If there does not exist a e V such that g,h c: Xa then we say that 
{g,h) is singular or (g,h) is a singular plane. 

It is clear that every plane in {X, V, /C) is either singular or nonsingular. Note 
that by Theorem 3.8.18, ifiY<Visa nonsingular two-dimensional subspace of V, 
then U^ is a uniquely determined nonsingular two-dimensional subspace of V. Now 
consider the following cases for two distinct intersecting lines g and h. 

Suppose that g is isotropic and h is isotropic. Then by Axiom T, there exists a 
unique a e M such that g,h cz Xa. If one ol g ov h is timelike, then by Axiom T, 
there exists a unique a e M such that g,h a Xa- 

Thus, if {g,h) is singular then either g is isotropic and h is spacelike, or g and h 
are both spacelike. 

Proposition 3.8.22. Let {g,h) be a singular plane with g isotropic and h spacelike. 
Then g L h. 

Proof: Let {P} =^ gnh. By Axiom S2, either g 1 h or for each A in g, there exists B 
in h such that P and APB are unjoinable. Suppose that A e g, B e h, and P and APB 
are unjoinable. It must be the case that A is joinable with APB; that is, gAjPB is 
nonisotropic. Let gPA = [6,e], so A,P\5,s. If A is unjoinable with APB then A, P, and 
APB are pairwise unjoinable points, so by Axiom 10, APB\b,z and APB e g. Thus, 
gPjPB "= gPA = g and B = PAAPB eg, so g = h. 

Thus, gpjPB ^ g and gpjPB is isotropic, so by Axiom T, there exists a unique 
y e M such that g,gpjPB c Xy. This means that P,A,APB\y so B = PAAPB\y, and 
h c Xy, which contradicts our initial assumption. Therefore, g ± h. ■ 
Theorem 3.8.23. Let {g{0),h(O)) be a two dimensional singular subspace ofV where 
g{0) is isotropic and h(0) is spacelike. Then {g(0),h(0))^ is a two dimensional 
singular subspace of V which contains giO). Moreover, i{g(0),h(0)}^)^ = (g(0),h(0)). 
Proof: First we observe that if W is a subspace of V generated by subspaces U and U' 



78 
then W^ = {U,U')^ = U^nU'^, v 6 {U,U')^ <^ v e U^, and 

V eU'^ -^v eU'^ n U^. Consider (g(0),h(0))^ = g(0)^ n h(Oy. Because g Ihhy 
Proposition 3.8.22 then there exists y,d e V such that (9 = y5 with g c Xy and 
h c .^6. From Theorem 3.8.20, giO)^ = <i(0),ts(0)) and by Theorem 3.8.19, 
h(Oy = </(0), Xy(0)) where / c .^5 is the unique line through O in .X§ orthogonal to 
/?. Hence, 

{g{0),hiO))^=g(0)^nh(0)^ 

= <^(0),-t6(0)) n</(0),ty(0)) 
= <^(0),/((9)>. 

Moreover, 

{g(0),kO))^ = g(0)^ nl(0)^ 

= (giOltsiO)) n{hiO),XyiO)) 
= (g(0),hiO)). 

Now if g,/ c .^e for some e e V then by Theorem 3.8.18, for O = sp, 
{giO)J(O)) = .ts(0) and <^0),/f(0)> = (g(0)JiO))^ = ^^0)^ = .tp(0). This says 
that g,h a Xp; that is, {g,h) is nonsingular. Thus, <g((9),/(0)> is nonsingular. 

If {g,h) is a singular plane, O e gnh, and g and /? are spacelike, then by 
Axiom SI, g is not orthogonal to h. So by Axiom 2, for all yi e g, there is a B e h 
such that O and AOB are unjoinable. ■ 
Theorem 3.8.24. For the above setup: 

1. go JOB e {g,h). 

2. go JOB -L g,/?. 

3. ifCeg andD e h such that go,coD is isotropic then go,coD = go job- 

4. (g{o),hio)y = {gojoB(o),giO)y. 

Proof: For 1., if there exists y eV such that gojOB,g c Xy, then a^,v405||y implies 
that B = 0^0 Iy and h c .Xy. This yields (g,h) - Xy is nonsingular. For 2. 



79 
go JOB -^ g,hhy Axiom S2 because by 1. above go,coD <= {g,h) and {g,h) is singular. 
lie & g and D e h such that go,cOD is isotropic then by l,.go,coD,gojOB c (g,h). If 
go,coD '^ go JOB then by Axiom T there exists a unique r\ e V such that 

g0,C0D, gOJOB <= -^y- 

Because two intersecting lines uniquely determine a plane, then (g,h) = ^Ey Hence, 
go,cOD = go JOB- The last conclusion follows from <^,/7> = (g, go job)- ■ 

If <g,/2> is a singular plane then from the proof of Proposition 3.8.22 and from 
Theorem 3.8.24, it follows that for each P e {g,h) there exists a unique isotropic line 
I (z {g,h) such that Pel; that is, {g{0),h{0)) contains one isotropic line. 

To complete the classification of orthogonal subspaces of V, it remains to 
consider hyperplanes. Now if {?i, V, /C) is any four dimensional affme space then a line 
and a plane which intersect in a point uniquely determine a hyperplane and any 
hyperplane in the space can be characterized as the subspace generated by a 
corresponding line and plane. 

Let A = </?, p> be the hyperplane generated by a line h and a plane p which 
intersect in a point O. Then 

From Theorem 3.8.19 and Theorem 3.8.20, we know that the dimension oi h(0)^ is 
three. By Theorems 3.8.18, 3.8.23, and 3.8.24 it follows that the dimension of 
(HO)^) = 2. Consider the following possibilities. If h(0)^ n $(0)^ = {!&}, then 
h(0)^ © p(Oy c V has dimension five whereas the dimension of V is four. If 
h(Oy n p(0)^ = p(0)^, then it follows that p(0)^ c hiO)^; h(0) 1 p(0)^ and 

KO) e (Hoy)^ = p(0). 

Which contradicts the assumption that h and p intersect only in O. 



80 
Therefore, h(0)^ n piO)^ - g{0) for some line g containing O. Because g must 
be either isotropic or nonisotropic, then the following is true. 
Theorem 3.8.25. IfA(0) js a hyperplane the there is a unique line g such that 
A{0)^ = giO) andiA(0)^y = A(0). ■ 

3.9 The Polarity 
In this section a polarity is defined so that we can obtain the metric through a 
process given by Baer [3|. For the convenience of the reader, the pertinent definitions 
and theorems [3] are given below. 

Definition 3.9.1. An autoduaJityn of the vector space V over the field /C is a 
correspondence with the following properties: 

1. Every subspace U of V is mapped onto a uniquely determined subspace n(U) of V. 

2. To every subspace W of V there exists one and only one subspace W of V such that 

7t(W) = U. 

3. For subspaces U and W of V, .U < W if, and only if, Ti(W) < 7r(ZY). 

In other words, an autoduality is a one-to-one monotone decreasing mapping of 
the totality of the subspaces of V onto the totality of the subspaces of V. 
Definition 3.9.2. An autoduality n of the vector space (V,/0 of dimension not less 
than two is called a poJarity, if n^ = 1, the identity. 

Definition 3.9.3. A semibilinear form over (V,/C) is a pair consisting of an 
anti-automorphism a of the field K, and a function y(x, y) with the following 
properties: 

(i) y(x, y) is, for every x, y e V, a uniquely determined number in /C. 
(ii) Xa + b, c) =y(a, c) +/b, c) and y(a, b + c) =./(a, b) +/(a, c), for a, b, c e V. 
(iii) y(/x, y) = (/(x, y) andy(x, /y) =./(x, y)a(0 for x, y e V and t e K. 
If a = 1 then /is called a bilinear form. 
Definition 3.9.4. If/is a semibilinear form over (V,/C) and if Z^ is a subset of V, then 



81 
{x e V : fix, u) = for everyu e U} and {x e V : J{u., x) = for every u e ^} are 
subspaces of V. We say that the autoduahty tt of (V, /C) upon itself is represented by 
the semibilinear form fix, y) if 

n{U) = {x s V -.fix^U) = 0} = {x e V :y(x, u) = for every u e U}. 

Theorem 3.9.5. [3] Autodualities of vector spaces of dimension not less than 3 are 
represented by semibilinear forms. ■ 

Theorem 3.9.6. [3] If the semibilinear forms f and g over (V,/C) represent the same 
autoduahty ofV, and if dim(y) > 2, then there exists aO ^ d e K. such that 

^x, y) = Xx, y)d for every x, y e V. ■ 

Definition 3.9.7. If n is an autoduahty of the vector space (V,/C), then a subspace W 

of V is caUed an N-subspace of V with respect to tt, if (v> <7i«v)) for every v e W. 

In this case, 7t is said to be a null system on the subspace W of V. 

Theorem 3.9.8. [3] Suppose that (f,a) represents the autoduahty n o/(V,/C). Then n is 

a null system on the subspace W <r^fiw,w) = for every w e W. M 

Definition 3.9.9. A fine <v) is called isotropic if (v) < 7i«v». (So an isotropic line is 

an N-line.) 

Theorem 3.9. 10. [3] If the semibilinear form (f,a) represents a polarity n, and if 

fiw,w) = 1 for some weV, then a^ = 1 and 

^{Kx,y)) =J(y,x) for every x,y eV. 

In this case we say that/ is a -symmetrical or Just symmetrical ■ 
Theorem 3.9. 11. [3] Suppose that n is an autoduahty of the vector space (V,]C) and 
that dimiV) > 3. Then n is a polarity if, and only if, % is either a null system or else n 
may be represented by a symmetrical semibilinear form {f, a) with involutorial a. ■ 
Theorem 3.9. 12. [3] Suppose that the polarity n of the vector space (V,/C) possess 



82 
isotropic lines and the dini(V) > 3. TJien n may be represented by bilinear forms if, 
and only if, 

(a) planes containing more than two isotropic lines are 'H-planes and 

(b) K, is commutative. ■ 

Theorem 3.9.13. Suppose that n is a polarity of the vector space (V,IC) such that the 
conditions of Theorem 9.12 are met. Then from Theorem 3.9.11 it follows that ifn is 
not a null system then n may be represented by symmetrical bilinear forms. ■ 

Defining the polarity n and obtaining the metric g. Consider (V,/C) 
constructed in this work. liU is any subspace of V then by Theorem 3.8.17, U^ is also 
a subspace of V. From the end of section 3.8, if W is a subspace of V, ZY 9^ {I©}, and 
U ^ V, then U-^ is a uniquely determined subspace of V. From Lemma 3.8.11 and 
Definition 3.8.5, {l©}-^ = V and V-^ = {I©}- So we define the mapping n on the 
subspaces of V as follows: 
Definition 3.9.14. If ZY is a subspace of V then tz{U) = U^ . 

From the remarks above and from Section 3.8 it is clear that to every subspace 
UoiV there exists one and only one subspace W of V such that Tt(>V) = U. 
Theorem 3.9.15. Let U and W be subspaces ofV, then U < W if, and only if, 
n(W) < n{U). 

Proof: Suppose that U < W. U OV e W-^ then OV 1 OW for every OW e W, U cz W, 
so OV 1 OU for every OU e U and OV e U^. Thus, ZY < >V implies that 
k(>V) = W^ <U^ = n{U). Conversely, suppose that W^ < U^. By Lemma 3.8.11 and 
Definition 3.8.5 we have ({1©}^)^ = {1©} and (V^)^ = V. From the previous section 
it follws that, if ZY is a nontrivial proper subspace of V then (U^)^ = U. because W^ 
and U^ are subspaces, then from the first part of the proof we have W^ < U^ implies 
that U = QA^)^ < (W^)^ = W. ■ 

Corollary 3.9.16. From Defmition 3.9.14 and Theorem 3.9.15 above it follows that 
n : U < V ^U^ <V is an autoduality. Moreover, because QA^)^ =U for every 



83 
subspaceU ofV, then % is a polarity. 

Lemma 3.9.17. Let^ e (& and OA e V. Then niU^) = n{U)^ for every 'i, e © and for 
every subspace U <V. That is, 7t is invariant under cj^ for every ^ e 0. 
Proof: We have 0{OAf- = 00^ A' = B e X, so (OA)' ^ OB e V. Hence, from Lemma 
3. 8. 6. the result follows. ■ 

Theorem 3.9.18. The polarity n is not a null system. 

Proof: Let a be any nonisotropic line with O e a. By Lemma 3.8.7, a is not 
orthogonal to itself so that d(0) $ d{0)^ and hence, d{0) $ n{d(0)). By Definition 
3.9.7, 71 is not a null system. ■ 

Theorem 3.9.19. The polarity n defined above may be represented by bilinear forms. 
Proof:. Let g be any isotropic line through O. By Definition 3.8.5, g{0) < giO)^ so 
that g{0) < nigiO)) and g{0) is an isotropic line in the sense of Definition 3.9.9. 
Conversely, if /z is a line such that h{0) < n{h(0)) = h{0)^ then h 1 h and by Lemma 
3.8.7, h must be isotropic in the sense of Section 3.2. Thus, (V,/C) possesses isotropic 
lines. By Proposition 3.7.3, dim(V) = 4 > 3. From Definition 3.8.21 any plane in 
(V, /C) is singular or nonsingular. Now a singular plane contains only one isotropic line 
by Theorem 3.8.24. If Z^ < V is a nonsingular plane then U = .ty(6>) for some y e V. li 
y e M then U has precisely two isotropic lines, as was shown in 3.2.2.8. If y e ^, 
then by Axiom 12 and Section 3.2, U does not have isotropic lines. Thus, no plane of 
V contains more than two isotropic lines. Since the field /C = R is commutative then 
by Theorem 3.8.12 the result follows. ■ 

Theorem 3.9.20. The polarity n may be represented by symmetrical bilinear forms; 
that is, there is a symmetrical bilinear form g such that for any subspace U ofV, 

U^ = n{U) = {OA eV: giOA^U) = 0} or 
g(OA, OB) = = giOB, OA)if, and only if, OA 1 OB for OA, OB e V. 

Proof: This follows directly from Theorems 3.9.18, 3.9.19, and 3.9.13. ■ 



84 



Thus, we have a metric g, a symmetric bihnear form, induced by our polarity, 
which agrees with our definition of orthogonal vectors, which in turn is induced by 
and is defined in terms of the commutation relations of the elements of our 
generating set Q. 

Lemma 3.9.21. Let g be a symmetric bilinear form representing n. Then g is 
nondegenerate. 

Proof: This follows from the fact that 7i(V) = V-'- = {l©}. ■ 

Theorem 3.9.22. Let g be a symmetric bilinear form representing n. Then there is a 
basis ofV such that the matrix ofg with respect to this basis has the form 



^ C ^ 
C 
C 



V 



-c 



^ C e IC: 



J 



that is, there is an orthogonal basis ofV such that the matrix ofg with respect to 
this basis has the above form. 

Proof: Let a e yW such that /C c .^a and let O = ap. Let 0|y,5 e Q such that 
a = 75 and y,5 1 p. Let s = yp e >! and r| = 5p e A^.Then zr\ = ypp5 = y6 = a so 
s 1 T). Put X = [a,y,5], y = [p,y,s], z = [p,Ti,5], and t = [a,s,Ti]. 

Claim. x,y,z,t are four mutually orthogonal nonisotropic lines through O. We 
have O = ap, O e x,t a Xa and O e y,z a Xp, so that xj 1 y,z. Also, 
O = aP = y5p = yr| = 58 so that y 1 t] and 6 1s. Then 

O e X cz Xy,0 e t a X^,0 e Xy, and O e z c Xr\ 

implies that x J. / and y ± z. 

The construction of the basis. Let E e fC, the multiplicative identity, 
considered as a point in .^a- Put T = EOE' e t and X = EOE'' e x. Note that because 



85 
O e x,t cz Xa and x 1 t then OxOi = gq = <7;a.r in Xa- Now t cz Xz and s e M, so 
there exist precisely two isotropic Unes, /Cig and ICje, through O in Xe. Thus, there is 
an unique Ee e /Cig such that T = EzOE'^. Because y c Xz then 7 s E^OEl e j; and 
CTya, = ao = CT/CTy in .^g. Similarly, because t cz X^, r\ e M, there are precisely two 
isotropic lines, K\^ and /C2T1, through O in .^ti and there is £,1 in K-\^ such that 
T = Er^OE'y^. because z c X^ then Z = ^nO^f^ e z and Gra, = ctq = a^a^ in X^. We 
calculate XOr = (EOE'')0(EOE') = (EOE)0{E'^ OE') = iEOE)O(EO'O'E') 

= {EOE)0{EOoE)' = (EOE)0(OEOOE)' = (EOE)00' 

= {EOE)00 = EOE e /C. 
YOT = iEzOEl)0{EzOEi) = {EzOEz)0(,EO OEz)' = E^OEz e /C,g. 
ZOr= (£^C'£--)0(£,iO£'^) = E^OEr^ e /C,n. Thus, if we put £, = OX, £2 = OY, 
£3 = OZ, and £4 = OT it follows that the set {Ei,£2,£2,£4} consists of four mutually 
orthogonal vectors such that £, + £4 is isotropic for / = 1,2,3. So if ^ is a symmetric 
bilinear form representing n then from ^, + £4 1 £, + £4 for / = 1 , 2, 3 and £/ 1 £j for 
/ ?iy it follows that for / = 1,2,3, 

O = g{Ei + E4,£i + £4) = g{£i,£i) + 2g{£i,£4) + g{£4,E4) = g{£i,£i) + g{£4,£4). 

So that giEi,£i) = -g(£4,£4) ^ O, because £4 is nonisotropic. Thus, it remains to 
show that {£\,£2,£3,£4} is a basis for V. But this follows from Section 3.7. ■ 
Theorem 3.9.23. (V,}C,g) js a four-dimensional Minkowski vector space. Moreover, 
(.ty(0),g-) is a four-dimensional Minkowski space for every y in Vq- 
Proof: Put g(E4,E4) = -1 and g(£„£,) = 1, for / = 1,2,3. Minkowski space is the only 
nonsingular real four-dimensional vector space with metric 



g 



f 1 








^ 





1 














I 





lo 








-1 J 



(3.24) 



86 
In the last section of Chapter 3 we show that each A, in ^ can be identified with 
a spaceUke plane and each cx e Q* with a reflection about a spacelike plane. 

3.10 Spacelike Planes and Their Relections 

For each X e g with Ol^i, define ax : V ^ V by axiOA) = (OA)^ = OA^, for 
OA e V. To extend this definition to any X, e ^, we note that if A. / O and OA e V, 
then there exists a unique D in X such that OO^A^ = D; that is, there is a unique D 
in X such that (OA)^ = O^A^ = OD. Thus, we define GxiOA) = OD, where 
OO^A^ = D. We note that if A | O then D = A^. 

First we show that 6x is a semilinear automorphism of V for each X e V. [23] 
Now, a function/: V i-> V is a semilinear automorphism if, and only if, it has the 
following properties: 

1. /is an automorphism of the additive group of V onto itself 

2. /sends one dimensional subspaces of V onto one dimensional subspaces (that is, / 

is a collineation). 

3. If A and B are linearly independent vectors of V, the vectors XA) and/B) are 
also linearly independent. 

Theorem 3.10.1. The map ax Js an automorphism of the additive group ofV onto 

itself. 

Proof: Suppose that ax{OA) = ax(OB). Then ax(OA) = OD where 

OD = OM^ = O'-B^. This implies that ^^ = 5^ or ^ = B. Thus, OA = OB and ax is 

injective. To show that ax is onto, let OB e V and A = OO^B^. Then 

axiOA) = 0^A^ = 0\OO^B^)^ = O^O^OB = OB. We show ax is additive. Let 

OA,OB e V. Let D = OO^A^, F = OO^B^. Then 

axiOA + OB) = (OAOB)^ = O^A^O^B^ = ODOF = ax{OA) + ax(OB). ■ 

Lemma 3.10.2. The map ax sends one dimensional subspaces ofV onto one 
dimensional subspaces ofV. 



87 
Proof: Let < C14 > be a one dimensional subspace of V. Then there exists a line 
g = [a, p,y] such that OB e< OA > if, and only if, 0,B e g. Since 
0,B I a,p,7 <^ 0^,5^ I a^,p^,y^ then d;,(< OA >) =< gx{OA) > is a one 
dimensional subspace of V. ■ 

Lemma 3.10.3. The transformation, di, maps linearly independent vectors ofV to 
linearly independent vectors. 

Proof: The vectors OA,OB e V are linearly independent if, and only if, goA '^ gOB if, 
and only if, go^A^ '^ So'^B^- Hence, &x is a semilinear automorphism of V for each 

x&g. m 

Theorem 3.10.4. The map ox : V ^^ V is a linear automorphism ofV onto V. 
Proof: Consider the definition of a semilinear automorphism [23,defn.73.11. Let (V,/C) 
and (V',/C') be vector spaces over the division rings K, and IC' , respectively. Suppose 
that ^i : /C H-+ /C' is an isomorphism from /C onto /C'. A map A : V i-> V' is called 
semilinear with respect to |x if 

1. X{A + 5) = X{A) + X(B) for all A,B e V. 

2. X(tA) = ^it)X(A) for all ^ e V and / e /C. 

The only isomorphism ^ : R >-► R is the identity. Thus, d;, is a linear 
automorphism of V onto V. ■ 

Let (V,/C,g) be a metric vector space. A similarity y of V is a linear 
automorphism of V for which there exists a nonzero r e )C such that 

g(yA,yB) - rg{A,B), for all ^,5 e V. If ^ is nonisotropic, r = ^^. The scalar r is 

called the square ratio of the similarity. 

Lemma 3.10.5. [23] A linear automorphism of a metric vector space V is a similarity 

if, and only if it preserves orthogonality. 

Proof: For all lines h and / in our space, h 1 I <r^ h^ 11^, hence, 

cxiOA) 1 diiOB) <^ OA ± OB. ■ 



88 
Theorem 3.10.6. The linear automorphism dx is an isometry. 

Proof: Let y : V i-^ V be a similarity with square ratio r ^ 0. Then for A,B e V we 
have g{yA,yE) = rgiA,B) = rg(y~\yA),y-\jB)) and giy-\yA),y-\yB)) = ]rg{yA,yB). 
That is, y"' has square ratio r"'. Now 6x is an involution, so that dx = ct^'. Hence, 
if A- 5^ is the square ratio of dx then r = -^ or r- = 1, so r = ±1. [23] Because the 
field K. is isomorphic to M, then V is not an Artinian space. Thus, for each similarity 
Ox of V there is an unique r > and there is an unique isometry a such that 
ux = M(P,r); where M{0,r)iA) = rA for all A in V. The map dx is thus a similarity 
with square ratio r^. Because r > 0, then from above, r = 1 and it follows that dx is 
an isometry. ■ 

Proposition 3.10.7. The isometry ax is a 180° rotation; that is, a reflection about a 
plane (a two-dimensional suhspace ofV). 

Proof: Let X & Q. Suppose that 0|?i and put O = Xa with a e M. Then for all A\X 
and for all B\a, dx(OA) = (OA)^ = O'^A^ = OA and 

Gx(OB) = (OB)^ = O^B^ = OB"^^ = OBO = -OB. 

Since ta(0) = tx{0)^ and V = .t^O) © tx{0)^ then d>. = l.t,(o) e -l.t,(o). Thus, 
ax is a reflection about the plane tx(,0). 

Suppose that O / A,. Let s | O such that s || X and let P | A. be arbitrary. 
Then if /? I s. FOR = Q \ X And OR = PQ. Thus, 

axiOR) = {ORY = (^0^ ^PQ = OR and ax V^^^o) = ^Uoy 

Now let PIO such that p 1 s, so O = sp. Let B \ p. Now pi?, because 8 || X and 
BOB = D\y where P = Xy and y || p. We calculate, 

ax(OB) = {OBY = {PDY = {PDy^ = {PDY ^ DP = BO ^ OB^ = -OB. 

Hence, ax = l.tj(O) © -1 ^^(^ji and ax is a reflection about a plane. ■ 



89 
To show that dx is a reflection about a spaceUke plane (EucUdean plane) for 
every X e ^, we use the following theorem from Snapper and Troyer [23]. 
Theorem 3.10.8. [23] Let {V,IC,g) be an n-dimensional metric vector space over a 
field K with metric g, K -'R, and n = 2. Every nonsingular real plane has a 
coordinate system such that the matrix of its metric is one of the following. 



' 1 , f 1 , 

- the Euclidean plane, - the Lorentz plane, and 

\ ^ I -1 ' 



-10., 

- the negative Euclidean plane. 
-1 ' 



Hence, the Euclidean plane, the Lorentz plane, and the negative Euclidean 
plane are the only nonisometric, nonsingular real planes. This also follows from 
Sylvester's Theorem [23] (It states that there are precisely n+1 nonisometric, 
nonsingular spaces of dimension n). 

Lemma 3.10.9. Let /C = R, n = 4, andV be Minkowski space. Then the orthogonal 
complement of a Lorentz plane is a Euclidean plane. 

Proof: Let {e,},=i 4 be a basis for V such that the metric of V with respect to this 

basis has matrix of the form (3.24). Let m = 63 + e4 and m = 63 - 64 so that 
< m >,< n > are the unique isotropic lines in the plane < 63,64 >. Let a be a Lorentz 
plane with isotropic basis mi and np Then there is an isometry cj :< 63,64 >•-► a 
such that a(m) = mj and (j(n) = n,. By the Witt Theorem ]23] a can be extended to 
an isometry of V, which we also denote by a. This implies that 

a : V =< 63,64 > e < 61,62 »-^ V = a ©< a(ei),a(62) >; 

that is, {CT(6i),a(e2)} is a basis for a-*-. Now cj is an isometry and 

^(ei,ei) = ^62,62) = 1, so the metric g with respect to a-^ has matrix I2, the 2x2 

identity matrix, and a-^ is a Euclidean plane. ■ 



90 
Theorem 3.10.10. The map a^ is a reflection about a spacelike (Euclidean) plane for 
every X e Q. 

Proof: From the proof of Proposition 3.10.7 it suffices to consider X e Q with 0\X. 
Let O = QX with 9 e A^ and let Q{') denote the quadratic form associated to the 
metric g obtained in Section 3.9. Because Q e M there are precisely two isotropic 
lines ACe,, ICq-, c Xq. Let O ^ A e ICq^ and O ^ B e ICq^. Then OA and OB are 
isotropic vectors which form a basis for ^e(O) and Q(OA) = = QiOB). Therefore the 
metric of ^9(0) with respect to OA and OB has matrix 



g = 



f Oa\, . f r 

(OA OB) = 
y OB j^ ^ ^ r 



where the products of the matrix elements are the inner products defined by the 
metric g and g{OA,OB) = r ^ 0. Since -L • O^ e /Ce , , and OB e ICq^ then /Ce,, ICq^ 
also form an isotropic basis for .te(0). Hence we may assume that r = \. Let 

OTi = -j^iOA - OB) and OZ, = -^{OA + OB). 

Then g'iOXx.OT,) = 0, Q'{OXx) = 1, and Q'{OT^) = -1. Because OX, 1 OTx and 
Ae(0) is nonsingular then 0X\ and OTx are linearly independent and hence, form a 
basis for .te(0). Moreover, the metric of .te(0) with respect to OJiTiand OF, has the 

'^''^^''' 1 -1 ]■ '^^^'^^^°''^' ^y Theorem 3.10.8, .te(0) is a Lorentz plane and by 
Lemma 3.10.9, .tx(0) = te(0)^ is a Euclidean plane. ■ 



CHAPTER 4 
AN EXAMPLE OF THE THREE-DIMENSIONAL MODEL 

This chapter begins by considering a net of von Neumann algebras, {7l{0)}oei, 
and a state co, coming from a finite component Wightman quantum field theory in 
three-dimensional Minkowski space. There are various senses to the phrase "coming 
from a Wightman quantum field theory". The assumption here is the version given by 
Bisognano and Wichmann [5]. That is, given a finite component Wightman quantum 
field, ^(x), assume that the quantum field operator, (j)(/), is essentially self-adjoint and 
its closure is affiliated with the algebra Tl(0) (in the sense of von Neumann algebras) 
for every test function /whose support lies in the spacetime region O. Driessler, 
Summers, and Wichmann show these conditions can be weakened [15]. But free boson 
field theories satisfy these conditions in three-dimensional Minkowski space [5]. 

For such theories the modular involutions, Jo, associated by Tomita-Takesaki 
theory to the vacuum state and local algebras of wedgelike regions, O, in three- 
dimensional Minkowski space, act like reflections about the spacehke edge of the 
wedge [5]. Since the modular involutions have that action upon the net, the 
hypotheses of Buchholz, Dreyer, Florig, and Summers (BDFS) are satisfied [6]. 
Therefore the Condition of Geometric Modular Action, CGMA, obtains for the set of 
wedgelike regions [27] in Minkowski space. The precise wording of this version of the 
CGMA is given below. 

Let /| and Ij be two lightlike linearly independent vectors belonging to the 
forward light cone in three-dimensional Minkowski space. The wedges are defined as 
the subsets WmJi] = {a/i + p/2 + /^ e R'-^ : a > 0, p < 0, (/^,/,) = 0, / = 1,2}, 
where ( , ) denotes the Minkowski inner product. 

91 



92 
Let 1 1 =(1,1,0) and 1 2 = (1,-1,0) be lightlike vectors in R''^, three-dimensional 
Minkowski space, and let V be the Poincare group, the isometry group of this space. 
Then the set of wedges, W, is given by W = {A.ff[/i,/2] : A. e V}, where 

XW[li,l2] = {Hx) : X 6 W[lil2]}. 

The CGMA for Minkowski space is defined as follows. Let {TZ(W)} fy^w be a net 
of von Neumann algebras acting on a Hilbert space 7i with common cyclic and 
separating vector Q e H, satisfying the abstract version of the CGMA and where the 
index set / is chosen to be the collection of wedgelike regions W in IR''^ defined as 
above. Recall from Chapter 1, with ({TZiW}} h'^\\,\'H,Q.) there is the following. 

1. A collection of modular involutions {Jw}iVeW- 

2. The group JT" generated by {JH'}ivew- 

3. A collection of involutory transformations on W, {'iw}weW- 

4. The group Tgenerated by {Tw}iVew- 
Assume also that: 

5. The group Tacts transitively upon the set W, that is, for every Wi,W2 e W there 

isaWi e yV such that iiv,ifV\) = W2. 

Note that this assumption is implied by the algebraic condition that the set 
{adJw}weW acts transitively upon the net {7^(^}jre>v- At this point the following 
two assumptions are added ([27] which have been verified for general Wightman 
fields). ■.•:•". 

4.6 For W\,W2 e W, if W\ nWj i/^ 0, then Q is cyclic and separating for 

'JZ(Wi)nn(W2). 

4.7 For W^, W2 e W, if Q is cyclic and separating for n{W\)r\ n{W2), then 
W\nW2* 0. 

The CGMA for Minkowski space is the abstract version of the CGMA with the 
choice of W for the index set /, together with assumptions 4.6 and 4.7 above less the 
transitivity assumption [6]. 



93 

Buchholz, Dreyer, Florig, and Summers [6] showed that with the above 
assumptions one can construct a subgroup ^ of the Poincare group P, which is 
isomorphic to Tand related to the group Tin the following way. For each t e Tthere 
exists an element g^ e ^ such that x{W) = g-^W = {gx(x) : x e ^}. To each of the 
defining involutions Tjy e 7T ^ e W, there is a unique corresponding g^y e ^c V [6]. 
Moreover, BDFS obtained the following (suitably modified for three dimensions and 
abbreviated for our purposes). 

Theorem 4.1 [6] Let the group T act transitively upon the set W of wedges in R''^, 
and let ^ be the corresponding subgroup ofV. Moreover, let gw be the corresponding 
involutive element ofV corresponding to the involution xiv e T. Then gw is a 
reflection about the spacelike orthogonal line which forms the edge of the wedge W. In 
particular, one has gwW = W, ' the causal complement ofW, for every W e W. In 
addition, ^ exactly equals the proper Poincare group V+. M 

Recall from Chapter 2 that the initial model of (G, 0) is as a group plane. This 
means that each g e Q is viewed as a line in a plane and each P - gh, g\h, as a point 
in a plane. Let us call the axiom system given in Chapter 2 as - A. Thus as- A is & 
set of axioms about "points" and "lines" in a "plane". 

Let P denote the collection of points P e 0. For each a e define the map 
aa : P u ^ ^P u ^ by 

^a{P) = ?"" = aPa-' for P e P and CTafe) = g"" = aga"' for g e g. 

Since {Q, 0) is an invariant system then each CTa is a bijective mapping of the set of 
points and the set of lines, each onto itself, which preserves the incidence and 
orthogonality relations, defined by "I", of the plane. We say that CTq is a motion of the 
group plane. Since Q generates then the set of line reflections g^ = {og : g e Q} 
generates the group of motions 0a = {cTa : a e 0}. Let O : -> 0<j be the map 
defined by 0(a) = aa, for a e 0. Then O is in fact a group isomorphism [2]. This 



94 
means that i'Q,<S>) is isomorphic to iGG,^G) (in the sense that Q is equivalent to Qa as 
sets and ^(Q) = Ga-, O(0) = ©a, where O is a group isomorphism). This implies that 
<^(as - A), which we denote by as - <t)(^), is an axiom system concerning the group of 
motions; line reflections of a plane, the group it generates, and point reflections of a 
plane. A plane whose points and lines satisfy as - A. 

As was shown in Chapter 2, given (^,(5) satisfying as - A, one obtains R*'-^, 
three-dimensional Minkowski space. Under the identifications given in Chapter 2, we 
find that each g e Q corresponds to a spacelike line in M''-^. Thus, as - <^(A) is a set 
of true statements concerning reflections about spacelike lines and the motions such 
reflections generate in three-dimensional Minkowski space. Moreover, since such 
motions are in fact isometries in R'--^ then 0(0) is isomorphic to a subgroup of the 
three-dimensional Poincare group. 

Theorem 4.2 Under the same conditions as in Theorem 4.1 it follows that 
{{\w}wey\>,'T) acting on W satisfies as- <i>(A). 

Proof: From Theorem 4.1 we have {{gw} w^w,"^) satisfies as - Q>i^A) since *p is the 
subgroup of V generated by reflections about spacelike lines. Also from Theorem 4.1, 
({tjf} WeW,T) is isomorphic to {{gw} WeW,^) so {{\w} w^y^,T) satisfies as - O(^). ■ 

The net continuity condition assumed by BDFS [6] for the next theorem was 
later shown to be superfluous [8] for this theorem and the remaining theorems. 
Theorem 4.3 [6] Assume the CGMA with the spacetimeR^-^ andW the described set 
of wedges. If J acts transitively upon the set {7^(fF)} ff'eW then there exists a strongly 
fanti-J continuous unitary representation U(P +) of the proper Poincare group which 
acts geometrically correctly upon the net {Tl{W)}weW and which satisfies U(gn) = Jw, 
for every W e W. Moreover, U{Vl) equals the subgroup of J consisting of all products 
of even numbers ofJw 's and J = U{Vl) u Jw^ U{Vl), where 
Wr^{x & M'-2 :xx > \xo\}. m 



95 
Theorem 4.4 Under the same hypotheses as Theorem 4.3, the group J is isomorphic 
toV + = ^, which is generated by the set of involutions {gw \ W e W}. Moreover, 
{{Jw }w&v^,J) satisfies as - O(^). 

Proof: By Proposition 1.1 there is surjective homomorphism "t^: J ~^T, where the 
kernel of ^, ker(Q, is contained in the center of J, Z{J). By Theorem 4.3 there is a 
faithful representation C/(P+) such that f/(gw) = Jw, for every W e W. Since the 
center of V+ is trivial. t/(') is a faithful representation of Vi and hence an injective 
map preserving the algebraic relations, Z{J) is trivial. This implies that ker(^)= {1} 
and ^ is an isomorphism. If T : T-> *p denotes the isomorphism of T and ^ given by 
BDFS from Theorem 4.1 then T o £, : J" ^ <p = 7?+ is an isomorphism. It therefore 
follows that the pair {{Jw] WeW,J) satisfies as - 0(A). ■ 

We can now give the main result of this chapter. 
Theorem 4.6 Any state and net of von Neumann algebras, coming from a (finite 
component) Wightman quantum field in three-dimensional Minkowski space, which 
satisfies the Wightman axioms, provides a set of modular involutions satisfying 
as-^{A). ■ 

As a final remark we note that since free boson field theories satisfy the 
Wightman axioms and therefore the CGMA for Minkowski space holds, then these 
theories give a concrete example of the three-dimensional case of this dissertation. 



CHAPTER 5 
CONCLUSION 

At this point it is useful to briefly recall the starting point of this thesis, to 
restate the problem, and to summarize the results obtained. It is assumed that there 
is a net of C*-algebras {Ajjiei, each of which is a subalgebra of a C*-algebra A, and a 
state CO on A. If this net and state satisfy the Condition of Geometric Modular Action, 
COMA, is it possible to determine the spacetime symmetries (the isometry group), 
the dimension of the spacetime, and even the spacetime itself, without any assumption 
about the dimension or the topology of the underlying spacetime? 

Recall also that the resolution of these questions involved two steps. First, 
given a set of involution elements G and the group © it generates, find necessary 
conditions on the pair (G, ©) that will allow a construction of three- and 
four-dimensional Minkowski space. Moreover, this should be done in such a way that 
the generating involutions can be identified with spacelike lines or spacelike planes 
and their respective reflections. This was done for three-dimensional Minkowski space 
in Chapter 2 and for four-dimensional Minkowski space in Chapter 3. 

The second step of this process is to determine what additional structure on the 
index set /would yield algebraic relations among the modular involutions, {J/},g/, 
such that the pair ({J/}/6/, J) satisfies the axiom systems given in Chapters 2 and 3. 
Using the work of Buchholz, Dreyer, Florig, and Summers [6] and the work of 
Wiesbrock [29], we are able to obtain a result concerning the above step in three 
dimensions. First we briefly recall the abstract version of the Condition of Geometric 
Modular Action, CGMA, described in Chapter 1. We assume there is a net, {7^,•},g/, 



96 



of von Neumann algebras acting on a Hilbert space H, where the index set I is a 
partially-ordered set. There is a vector Q e H which is cyclic and separating for each 
TZi, i e /. From the modular theory of Tomita-Takesaki we then obtain a collection, 
{J,}, -6/, of modular involutions which generates a group J" and a collection, {A,}/g/, of 
modular operators. The assignment i ^ TZj is an order-preserving bijection and each 
ad Jj leaves the set {7^,}/e/ invariant. The last two assumptions imply that for each 
/■ € /, there is an order-preserving bijection t, on / such that J,7?.yJ, = 7^t,(/> J e /. 
The group generated by {t,}/6/ is denoted by Tand forms a subgroup of the 
transformations on the index set /. Assume also that the two intersection conditions 
for wedges given in Chapter 4 also hold as part of the CGMA for what follows. 
To help explain the additional assumptions used in our result we give the 
following definitions and theorems. 

Definition 5.1 [6] The Modular Stability Condition (MSC). Let {'ll{W)]w^w be a net 
of von Neumann algebras satisfying the CGMA where the index set / is the set of 
wedgelike regions W in E'-^ described in Chapter 4. Then the modular stability 
condition is satisfied if the modular unitaries are contained in the group J" generated 
by the modular involutions; that is, 

A'^ 6 J for alU e M and W e W. 

Theorem 5.2 [6] Assume the CGMA for three-dimensional Minkowski space with (4.6) 
and (4.7), where the index set / = W, the collection of wedgelike regions in i?''2. 
Assume also the transitivity of the adjoint action of J on the net {n(W)} ^.^vv- Let 
f/(R'-2) be the representation of the translation group. IfA'^ e J, for all t e M and 
some W e W, that is, if the modular stability condition obtains, then sp(U) c V+ or 
sp(JJ) d V-. Moreover, for every future-directed lightlike vector H such that 
W+lczW, there holds the relation A'^U(t)A-^' = f/(e-«'f), for all t e R, where 
a = ±271. 

97 



Definition 5.3 [29] Let J\f, M be von Neumann algebras acting on a Hilbert Space 
Ti-Let Q be a common cyclic and separating vector in H. If A^' ATA^ c A/", for all 
/ > 0, we call (M d M,Q) a +-half-sided modular inclusion (+-hsm). If Ajt( A/" 
A'}!^ d jV, for all r < 0, we call (Af c: M,Q.) a —half-sided modular inclusion (— hsm). 
Definition 5.4 [29] Let J\f, M be von Neumann algebras acting on a Hilbert space TC 
with Q e H a common cyclic and separating vector for A/", M., and A/'n Ai. 

1. If ((AAn M) e A/',Q) and ((A/'n A^) c A/(,Q) are ±-hsm inclusions. 

2. And if J^is - lim,^+oo A'ifA~/\)JAf = s- lim,-^+oo A'j^Aji'. 

Then we say that such a pair ((M, M), Q) has (±) modular intersection, ± mis. 
Theorem 5.5 [29] Let J\f, M, C, and U be von Neumann algebras acting on a Hilbert 
space H with a common cyclic and separating vector Q e Ti. Assume the following. 

I. l.(M,M,n) is- mis, 

2. (£,M,D) is+ mis, 

3. (£,A/',Q) is- mis, 

II. l.iMdMQ)is-hsm, 

2. AdJM(Jj{fJM) = JMJj{f, 

3. [AdJc{JuJM),Ju-^Mh 0' 

III. l.Ad (AdJciJj^JNY) {Ad A^V/;(JaA/^))(J^J^)2(A0 c M, 
with t a = ^ln2. 

Then the modular groups 

A'ir, A'jii, A'l, and A'j^, for t, r, s, v eR, 
generate a unitary representation of the 2+1-dimensional Poincare group. ■ 
Remarks [29] The conditions in I. give a unitary representation of the 
2+1-dimensional homogenous Lorentz group. The hsm inclusion of condition II. equips 
us with a representation of the translations along some light ray. The product of the 
two modular conjugations is then a finite translation of this kind. Moreover, due to 



98 



the result of Bisognano and Wichmann [5], the modular conjugations of the wedge 
algebras act as reflections. These properties are encoded in condition II. 2 and II. 3. 

A physical framework will now be given as a description of quantum field 
theories in terms of local nets of algebras [17]. The basic assumptions are the 
following. Let {A(0)}oeV c B(H) be a net of von Neumann algebras indexed by the 
closed double cones V in M*'^ which satisfy the following properties: 

1. (Isotony) If O, c Oj, then A(O0 c AiOi). 

2. (Locality) If Oi c Oj, then AiO\) c AiOi)' . 

Where AiO)' denotes the commutant of A(0) in B(H) and O' is the causal 
complement of O c R''^. 

3. (Poincare covariance) There is a unitary representation 

U : SO (2, 1) > R'-^ -> U(H) of the Poincare group with positive energy. 

4. (Vacuum vector) There is a unique W -invariant vector Q e H. 

The algebra AiO), the inductive limit of the net, is called the local algebra of 
observables localized in O c R''-^. 

As was mentioned in Chapter 4, if the local net is generated by Wightman 
fields then the modular groups associated with algebras of observables localized in 
wedges act as Lorentz boosts in the directions of different wedges and the modular 
conjugations act as reflections [5]. 

In particular, the adjoint action of the modular conjugations on the net act as 
reflections about the spacelike edge of the wedge. For what follows we shall call the 
properties in the previous paragraph the Bisognano and Wichmann property. 
Theorem 5.6 [29] Let AiO), O c R''2, be a local net fulfilling the Bisognano and 
Wichmann property for wedges. Let M = A^/1,/2]), M. = A^/1,/3]), 
£ = AiW\li,h'\), andk^ AiWlU^h,-^-^, where hJj, and 1 3 are three linearly 
independent light rays. Then this set of algebras together with the vacuum vector Q 
fulfill the assumptions of Theorem 5.4. Conversely, let M M, £., and M be a set of A 



99 



VOD Neumann algebras acting on a Hilbert space H together with a common cyclic 
and separating vector Q. e H, which fulfill conditions I.-III. of Theorem h A. Then 
these data determine a local (Bisognano-Wichmann) net A{0) e B{H), O c R'''^, 
such that the incident algebras become the wedge algebras of the constructed net as 
in the first part. ■ 

Theorem 5.7 Let {Tli} j^i be a net of von Neumann algebras acting on a Hilbert space 
^, together with a common cyclic and separating vector Q e H satisfying the CGMA 
with the MSC. Assume also that there exist iM,JM,kj{f,lc e / such that 
'^w^'^jM^'^kj,', andUi^ satisfy the hypotheses of Theorem 5.5. Then the following are 
true. 

1. There is an injective map F : W ^I, such that for each W e W, the modular 
objects J F(W), ^'f(W) have the Bisognano-Wichmann Property when acting upon 

'R.F(W)- 

2. The modular unitaries A}'^,, A/^, AJ^^^, and A)^ generate a continuous unitary 
representation of 7^+ which acts covariantly upon {TIf(W)}w^w- 

If the additional intersection assumptions, (4.6) and (4.7), are made on the subnet 

{TZfifV)} iVeW, then: 

3. The CGMA as stated for Minkowski space holds for the subnet {TIf(W)} w^w, as 
does the MSC. 

4. The modular conjugations { adlpov) } w^m and thus {{xf(W) } We w, T), satisfy the 
axioms of Chapter 2. 

5. There is a continuous (anti-) unitary representation of P^ acting covariantly upon 

Proof: For the proof of Teorem 5.7.1, we give the construction Wiesbrock gave in the 
proof of Theorem 5.6 [29]. Let /, = (1,1,0), h = (1,-1,0), /j = (1,0,1) e R''2. The 
local algebra of observables to wedges is defined by 

For arbitrary linearly independent light rays /,,/,■ e M'-2 pointing to the future, let 



100 



Al^j € iSO' (1,2) with // = A/^./ 1\, and Ij = Aj.j.lj. This element in S0^(\,2) is 
uniquely defined up to a multiplication by a boost of type Ai^j^{t), t e M with the 
given asymptotics l\,l2- Let U denote the unitary representation of the Poincare group 
according to Theorem 5.4, that is, let 

ZY(A/„/,(0) -Af;, U(Ai,i,it)) ^ A'ijl, U{A,^j^(t)) ^ A^iYc ' ^ ^- 

Now define the observable algebra associated with arbitrary wedges by 

n{W{liJj-\) = adU{Ai^j){Jlij^) ( e {7^,}/e/ by the MSC and the CGMA). 

For translated wedges, define for a e R''-^ 

n{W[liJj,a\) = adU{a)U{A,^j){nij^) ( e {Te;}/^/ by the MSC and CGMA). 

In this way, for any wedge region PF in R''^ there is a unique von Neumann algebra 
Uj^ in {7^,},•e/. Taking F : W ^ I to be the map F{W) = />,, iox W e W and /> e / 
as obtained above and the result follows. 

Conclusion 2 follows from Theorem 5.6 and Theorem 5.5. Conclusions 3 and 4 
also follow from Theorem 5.6. The last conclusion follows from Theorem 4.3. ■ 

We conclude this chapter with a few remarks. Given a net {^,},e/ and a state 
to satisfying purely algebraic conditions, one derives three-dimensional Minkowski 
space, and an identification between elements of / and the wedges in 
three-dimensional Minkowski space in such a way that the adJi act like a reflection 
through the spacelike edge of the wedge. Therefore, solely with assumptions on the 
algebras of observables {^,},g/ and the preparation co, we are able to derive the 
physical spacetime and its symmetries. We can also derive an interpretation of 
suitable elements of {^,}/e/ as local algebras associated with wedge regions, as well as 
derive a prescription of how the spacetime symmetries act upon the observables. In 
addition, we can get a time orientation of the spacetime from the MSC. 

101 



A similar process can be done for four-dimensional Minkowski space using the 
work of Wiesbrock and Kahler [18]. However, we refrain from giving the details here. 



-'■ -t '-'■': r.- 



102 



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103 



104 

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105 



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[31] J.W. Young, Projective geometry, MAA, Chicago, 111., 1930. 



BIOGRAPHICAL SKETCH 

Richard K. White was born in Richmond, Virginia on August 19, 1960. He 
graduated summa cum laude from the University of North Florida in 1991 with a 
Bachelor of Science degree in Mathematics. He graduated from the University of 
Florida in 1994 with a Master of Science degree in Mathematics. In May 2001 he 
graduated with a Ph.D. in Mathematics from the University of Florida. He is the 
proud parent of a six-year-old angel, Jackie. 



> * . ' i**^ i 






106 






I certify that I have read this study and that in my opinion it conforms to acceptable 
standards of scholarly presentation and is fully adequate, in scope and quality, as a 
dissertation for the degree of Doctor of Philosophy. 

Stephen J. Summers. Chairman 



Professor of Mathematics 



1 certify that I have read this study and that in my opinion it conforms to acceptable 
standards of scholarly presentation and is fully adequate, in scope and quality, as a 
dissertation for the degree of Doctor of Philosophy. 



Gerard Emch 

Professor of Mathematics 



I certify that I have read this study and that in my opinion it conforms to acceptable 
standards of scholarly presentation and is fully adequate, in scope and quality, as a 
dissertation for the degree of Doctor of Philosophy. 



/^ . f-yC^'-^-JU^ 



JohA R. Klauder 

Joint Professor of Mathematics and Physics 

I certify that 1 have read this study and that in my opinion it conforms to acceptable 
standards of scholarly presentation and is fully adequate, in scope and quality, as a 
dissertation for the degree of Doctor of Philosophy. 



Itc^ /h^-,. 



David Groisser 

Associate Professor of Mathematics 

I certify that I have read this study and that in my opinion it conforms to acceptable 
standards of scholarly presentation and is fully adequate, in scope and qualitv. as a 
dissertation for the degree of Doctor of Philosophy. 



OL^\LA^h 



Khandkera Muttalib 
Professor of Physics 

This dissertation was submitted to the Graduate Faculty of the Department of 
Mathematics in the College of Liberal Arts and Sciences and to the Graduate School and was 
accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. 

May 2001 



Dean, Graduate School 



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Uj^bSl- 



UNIVERSITY OF FLORIDA 



3 1262 08555 1967 






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